The Poles are making a very risky gamble because while the UK isn't important to Poland's strategy, it's become clear to them that France isn't prepare to do much of anything without British aid. So they are:
1) Playing for time. Hopefully circumstances change for the better before they need to commit to anything drastic. Their efforts to get Japan to shift its aggression from the UK to the USSR were in service to this.
2) Positioning themselves as a leading member of the Axis through initiatives such as the Lwow Wargames. Hitler's expansions thus far have all been contingent on the support of other states, befriending Germany's nominal allies will likely restrain Hitler's demands. Already they've they've established a strong partnership with Hungary.
3) Poking around and see if there's any threads to pull on. Hopefully there are groups within Germany that may restrain German territorial ambition. Which is to say that it was no accident that the virtues of the Colonel's Regime ended up being a topic of discussion at Lwow.
4) Learning how Germany intends to fight and planning accordingly.
It's admittedly out of character for Beck, but finding a way out of this nightmare scenario requires some creative decisions.
Well most of that could be achieved though without having German troops participate in war games
in Poland (besides which it would be quite likely that many in the Polish military would have massive misgivings about a German corp being able to drive through Poland, scope out the terrain and basically getting a test run of how things might go when they want to roll into Poland). Having war games in say Hungary or Romania would achieve most of that. Poland could still have the initiative by suggesting them in the first place and assisting in organizing them with its ally (Romania) or new friend (Hungary).
[EDIT: Also, given how soon the POD was and how things generally were in the 1930s, it would probably be likely that instead of military exercises in which Poland and Germany participate, it might be more of a case where Poland suggests military exercises to be hosted in say Romania at which German and Hungarian observers are invited and then perhaps suggest that Polish and Romanian observers could be present for any military exercises hosted by Hungary and involving Germany - this would be aimed at building up towards cultivating relationships with Germany's nominal allies, decreasing the heat in the German-Polish relationship and perhaps directing Germany towards Ukraine via Hungary/Romania]
[EDIT 2: Also given Stalin's paranoia, it is quite likely that he would (correctly) interpret any such moves as a possible coming together of Germany and the Central European states to collude in attacking the USSR even if the truth is a bit more nuanced insofar as Poland might not want to directly attack the USSR but is basically attempting to deflect German attention away from itself and onto its neighbours that it also just doesn't happen to be chummy-chummy with such as Czechoslovakia and the USSR. In Moscow, Stalin might well wonder if there is a secret annex or protocol allowing Germany to cross over Polish lands in order to attack the USSR. That said, Stalin would not be sitting still. He would likely also be trying to get a new agreement with the French and as in OTL attempt to get a deal with the Germans, however here one of the driving factors would be to undermine and outdo the Poles in their attempt to deflect German attention on their enemies and get it directed back at Poland. This would I imagine, be a success as in OTL though the timing might likely vary. The reason of course is as
@Polish Eagle pointed out, that the most direct route from Berlin to Moscow lies through Warsaw (and not Bucharest), so Hitler is eventually going want to pressure Poland into allowing this much as happened with the Yugoslav-Axis negotiations and these negotiations are either going to fail or result in the overthrow of any Polish government that agrees to them and once Germany invades Poland it isn't going to establish a puppet regime within the pre-invasion borders, it is going to occupy all of Poland and annex western Poland. Agreeing to let the USSR then occupy eastern Poland will make sense in that it gives Germany a chance to fend off France, if indeed France still honours the Franco-Polish alliance and also making it impossible for the USSR to become immediately allied to France and Poland if the French (in the face of British distraction to the war in Asia) insist on the Poles agreeing to the Soviets joining as allies and yielding everything east of the Curzon line as a price to save the rest of Poland - since of course the USSR would become an accomplice in occupying Poland. Of course if things develop more fluidly, Stalin may not get to secure an agreement with Germany concerning Poland before Germany decides to pull a Yugoslavia type scenario on Poland and invade as a result of failed negotiations or an overthrown government, in which case events might move rapidly and the French might now succeed in 1939/1940 where they failed in 1935/1938 with Czechoslovakia and secure Polish agreement to Soviet involvement and intervention and they might do so precisely because the French might be reluctant to act without the British otherwise and might then wish to secure an alliance involving Poland, the USSR, Yugoslavia and Romania against Germany.
Of course, any war in Europe in this TL might well end up similarly to what happened in
@pdf27 's also superb TL,
A Blunted Sickle, except here the British have more combat experience with the Japanese. Now that would be an interesting cross-over of TLs...]
In fact, staging them in Romania might form part of a strategy to encourage German attention to be directed to the USSR (specifically Ukraine) by way of going through Hungary and Romania rather than Poland. In this regard I would expect any agreement that Poland ever signs (which seems rather doubtful) to be similar to what Yugoslavia tried to wrangle out of Germany in OTL 1941 where Germany publicly agrees it would not allow troops to cross Yugoslav territory and that it wouldn't use the Yugoslav rail system for that (though Yugoslavia also had included war materials in the agreement but Germany's announcement excluded that indicating that Germany was at least hoping to use the Yugoslav rail system to transport war materials to the Greek front).
Additionally befriending Hitler's allies in an attempt to restrain Hitler's demands, whilst a good idea is rather undermined by actually entering into a client relationship
with Hitler. The former idea is pretty much what the French and British tried in trying pry states like Italy away from Germany and it is certainly a creative idea insofar as it would mean Poland trying to befriend Hungary and shore up its alliance with Romania and Yugoslavia. In this way it would be a way of isolating Hitlerite Germany, which in its own way buys time since a Germany that doesn't have Hungary or Romania fully onboard cannot surround and invade Poland more easily (and might likely not get any support from them either in any disputes that may arise). Similarly it might mean Poland reaching out to Italy.
I'm just not seeing Beck or the Polish leadership in general going so far as to join the Axis and Germany formally and becoming a client state because even in OTL's nightmare scenario of the post-Munich crisis elucidated by Hitler's dismemberment of Czechoslovakia, Poland maintained a solid stance of spurning any kind of security guarantee with the USSR which both the UK and France were looking to obtain to really put pressure on German (and this despite now open German threats against Poland). In that nightmare scenario, creative decisions were required but they weren't really forthcoming.
EDIT: Also in regards to France not acting without Britain, there would be three things also to consider:
1. Even if France strongly indicates a reluctance to act without the UK, this is still materially different than France repudiating its military alliance with Poland. In the former scenario, Poland still derives a modest benefit insofar as Germany is unlikely to send 100% of its forces against Poland if there is still a chance of French intervention. Obviously Germany would still feel confident enough to send large numbers of forces against Poland, but it might retain more forces to defend its western border than it otherwise would have if France openly repudiated its alliance with Poland and basically indicated Germany had free reign to do whatever they wanted to Poland.
2. France was not actually ever obligated to to send troops to Poland to aid in its defence or to declare war on Germany until the
new agreements of May 1939. Prior to that the Franco-Polish alliance from the 1920s had the aim of France keeping lines of communication open and assisting Poland as it saw fit and trying to keep Germany in check prior to hostilities (and perhaps even during hostilities with the threat of intervention).
3. The UK never obligated itself to even assist Poland if it was attacked until March 1939 with the Anglo-Polish Agreement.
This would suggest that Polish foreign policy in 1938 would tend to continue in a similar vein as from 1923 until 1937 when during that time:
- the UK wasn't obliged to assist Poland (but would assist
France if Germany attacked France)
- The French weren't even obligated to declare war or send troops to assist Poland if Germany attacked it.
The situation outlined thus far would also strongly seem to favour the Poles desperately trying to act as peacemakers between the Japanese and the British in order to allow the Japanese to be able to focus more on the USSR and to allow the UK to refocus on Europe. This would also likely mean the Poles might try to encourage a settlement between Hungary on the one hand and Romania and Yugoslavia on the other hand. Perhaps by sponsoring talks on border issues and minorities. Because if they can somehow get Hungary, Romania and Yugoslavia to bury the hatchet then they remove a reason for Hungary to throw in its lot with Germany and can aim to build a Central European alliance network of Poland, Hungary, Romania and Yugoslavia (and perhaps Bulgaria if they assist in similar talks involving Bulgaria and Romania/Yugoslavia) and maybe Greece and link it with the alliance with France.
Like Benes did after the Rhineland militarization, Beck is likely to read the tea leaves just as well and realize that abandoning the alliance with France and simply coming to an accommodation with Germany would mean the end of his country's independence