Philip was already making his claim to the English throne when the Armada set out.
I didn't find any credible mention of that (sure, a lot of "Spain was gonna invade and Inquisition was going to kill everyone for Phillip crowning himself as king").
After the death of Mary, he lacked any real ground for claiming anything, and basing himself on the respect of traditional and catholic institutions, couldn't pull a Paradox game move to take-over England.
Does that make the intelligence of Spanish command in this matter clear?
I understand Spanish Dark Legend still have its amators, but let's consider them as able of basic reflexion as their opponents, shall we?
First, a bit of context would help. Remember that Philipp not only tolerated Elizabeth as an anglican queen (mostly in order to avoid a Valois presence in Britain and more generally in Atlantic) but actively tried to prevent the Roman Church to pursue an aggressive stance.
The relations between Spain and England worsened essentially due to the English policy towards Netherlands (namely supporting the revolt).
It's relativly safe to assume that in face of its main geopolitical goals (Ottomans in Mediterranea and Dutch in Europe) as highlited for example by the Treaty of Joinville, critically with the French situation becoming more hard to hold with the political establishment of Bourbons - even before the death of Henri III - Philipp II would turn to his usual English policy : prevent the appearance of a northern and atlantic European entente.
Depending on how the Spanish expedition fare, and how politically skilled could be the commanders. Personally, I think that Alexander Farnese actually taking care of the situation would be needed.
The man prooved to be prone to compromise on strategical level.
The main goal, again, being the repression of Dutch revolt, I would rather see him either letting Elizabeth I on throne (it helped that she was far less enthusiast on the war, seeing Netherlands more as a political lever than a goal in her negociations with Spain) under several conditions (such as naval limitations and imposing a successor), or (depending on the pressure made on the expedition and policy of Elizabeth) choosing another king while it may proove hard giving the rarity of clear candidates, rather than loose time and ressources in face of Dutch and French fronts.
People as Medina Sidona could be, on this regard, more prone to comply before pressure (while not ourtight ignoring Philipp's interests, but the king didn't had a clear plan on it, and would have focused more on different fronts) and to be less efficient, both on long and short term (maybe a second Spanish-English war in the 90's)
Also, regarding that faction seeking to limit royal power--such a thing was stirring into existence at this time. Among the Puritans.
So good luck to Spain signing them on.
As surprising it might be, Puritains and Court weren't the only groups in existance as would be two poles of a magnet.
Not only such groups were heterogenous (Elizabeth being far more cautious than her council, or Puritains being divided) but more informal "interest groups" (as we could call them nowadays) could be called if not created outright (as Philip heavily supported what were at first several catholic movements to become the powerful Ligue).
I would tend to think, depending on the skills of Spanish responsible, that the rural "middle-class" and the high aristocracy may be the main beneficiaries, as well a good part of maritime middle-class trying to compromise to avoid having their naval edge being damaged too importantly.
(At this point it's a matter of these groups managing to compromise between themselves in a first place, maybe using the prestige of a noble outside the Elizabethan council, and then both Spaniards and these English to reach a quick agreement).
As an aside, you misunderstood and partially ignored the point : reject of both royal and parlementarian power. Basically reject of hegemonic powers gaining more political reach that they traditionally had. Without turning to pre-Tudor situation (something that would have been impossible anyway), I could rather see one or different groups working on this base and while failing to stop the process of bureaucratisation and hegemonic power, at least try to temper it by political "lobbying".
I'm not talking about an half-millenarist/religiously based political stance, or reject of Elizabeth as such, but reject of personal royal power (I should have precised, my bad) that was the subject of much criticism since the 70's in the Parliment (as the refusal of the queen to marry or designate an heir prooved).