Comparing acidic strength in oxan-2-one and oxan-4-one
This has a simpler answer than you expect. First of all, your reasoning is wrong because inductive effect is much less dominant than conjugation of electron pairs. So, if the latter is unequal in both cases, we should first prioritize it over inductive effects.


  Removing a proton(from the carbon adjacent to the pi bond) would lead to conjugation


It seems you already know that the beta hydrogen of carbonyls is most acidic. Both molecules have carbonyl groups, and both have beta acidic hydrogens. So, you're on the right track!

Whichever molecule is more acidic, it will have a more stable conjugate base.

Note that in the second anion, the excess negative charge is completely delocalised over to the carbonyl group. Compare that with the first anion, where both the excess negative charge and the oxygen's lone pair are delocalising with the same carbonyl group. Thus, the excess negative charge is unable to completely delocalise with the carbonyl group.

Can you now identify why the first molecule is less acidic?