This would be a mess but I do see some opportunities to resolve the crisis as a domestic affair of the HRE in a way that was impossible OTL. OTL Rudolf dithered until well after the last Duke died while the two protestant claimants essentially went in and occupied the duchies to claim their inheritances. OTL they did that in part because the Protestant Union secured the backing of Henry IV. But now Henry IV may be pushing a different claimant. I don't think that gives Rudolf any impulse to suddenly favor Neuburg or Brandenburg, he's probably just as likely to make a sudden rash decision like he did OTL (briefly picking the distant Saxon claimant before being forced to back down). But if he's smarter (or gets better advice) it may give him more leeway in pushing to sequester the Duchies. OTL this was seen as a contravention of the two protestant princes rights and French intervention was cast as a king of mediation. But here French intervention would be seen as foreign interference in favor of a French claimant over the German ones so the German princes may be more willing to entertain a sequestration and regency in the Duchies while the succession is adjudicated in the Imperial courts. Spain would probably have more reason to get involved than it did OTL but it's still hamstrung by lack of men and money and the precarious Twelve Years Truce with the Dutch, only recently signed.
A lot depends on
1. What is the religion of de la Marck. I'm not sure in what faith he was raised OTL and of course he could always convert if he had lived longer. If he's a Calvinist that makes things much messier for Henry as far as internal French politics go but may make him more acceptable within the Duchies. Cleves was Lutheran but trending Calvinist due to Dutch influence, Julich was mostly Catholic but Mark and Berg where a mix of Catholic and Lutheran.
2. If Rudolf pushes sequestration does he still appoint his cousin Leopold as Imperial envoy (a bad move OTL), the nature of any regency probably dictates how the German princes react to it.
3. What is the position on the Dutch? OTL they sided with the French in supporting the Protestant Union. A protestant de la Marck puts them in a real bind while they certainly would be unwilling to support a Catholic de la Marck. I think of the two Germans they favored Brandenburg but like the rest of the Protestant Union would have favored an amicable settlement between the the two to preserve the unity of the Protestant cause. But with the recent signing of the Twelve Years Truce and caught between the candidate of France and the German protestants the Dutch may see it as a no win scenario and avoid intervening. This takes some pressure off of Albert and with essentially no foreign support perhaps the Protestant claimants are more amenable to Imperial mediation.
4. What is the position of England? Ditto much of the concerns the Dutch face. Almost certainly they just stay on the sidelines, with James hoping to fill the role of honest broker.
5. Is Henry IV still assassinated or does he live longer to direct French intervention. If he dies I'm not sure that France would go to bat for de la Marck, OTL the French did still invade OTL but only briefly to assist the Protestant Union in ejecting Leopold from Julich and once that was done they got out quickly. So de la Marck's claim evaporate if Henry dies. But an assassination may be less likely if de la Marck is Catholic because in this case Henry wouldn't be backing a German protestant but a French Catholic. So the French Catholics may not get so riled up about Henry moving against Spain/Catholicism/the Pope if the claimants are a French Catholic vs a couple of German Protestants.
While the internal politics with a Catholic de la Marck are ironically easier for Henry (and thus make French intervention all the more likely) they ironically provide and opportunity for the Emperor as well in so much as OTL Henry kind of stepped into the void left by Rudolf's failure to expeditiously handle the matter and assumed a role of arbiter, basically usurping the mantle of authority that by rights should have been the Emperor's. I read something a while back that referenced Archduke Albert in particular being pretty concerned by this. Being, I think, a more savvy ruler than his brother, he was very conscious of the damage this could cause the Emperor and was in particular most concerned that the Emperor's authority be respected by all claimants. More it seems than he was with securing the neighboring Duchies for a friendly Catholic prince (perhaps because that was nigh impossible). So in a way Henry backing a French candidate could give a capable Habsburg more room to maneuver in resolving the crisis. To the extent to which the succession conflict becomes confessionalized it also becomes nationalized in so much as the Catholic claimant is a foreign French prince while the protestants are German. This still holds to a lesser extent with the French claimant being Calvinist as he wouldn't be covered by the Peace of Augsburg and he'd of course still be in opposition to the German protestants. So if the Emperor could pivot that way he could use the anti-French angle as cover for an decision on the succession in such a way as to placate both the Catholic and Protestant factions within Germany. This would be made even easier if Neuburg's own son and heir Wolfgang Wilhelm completes his OTL conversion to Catholicism a little earlier than OTL. But I doubt Rudolf is the man to thread that needle.
Supposing the crisis does spiral out of control into war and Henry pushes to widen that war by building on his recent alliance with Savoy, and Spain is dragged in, to secure Northern Italy, the Spanish Road and their flanks in the Netherlands then the Dutch may possibly follow suit and the Twelve Years Truce could collapse early. i don't know if this means an early 30YW because the sides are stacked very differently with the French possibly opposed to the protestants and the Imperial succession not yet secured (Matthias had still not been named Rudolf's heir at this point and OTL Ferdinand II is even father down the totem pole). If it goes badly, and it easily could, it may quash any notion of French supporting Gonzague in Mantua down the road. At the very list this completely changes the course of French foreign policy in the 17th century. The precedent set by Henry IV of favoring the German protestants to further French interests is gone, instead French policy is built around breaking the Spanish Habsburg encirclement through alliances with neighboring powers, like Savoy, and the extension of French power through the installation of French clients, like de la Marck in Julich and Gonzague in Mantua. They'd make for a much less appealing partner to the Protestant Union in Germany or to England for that matter.