I've never had problems with bruising on the palms unfortunately.
Curious to see what your trainer thinks though!
I do like the fingerless gloves sometimes. Some of the barbells, dumbbells, handles for machines, etc at my gym have like a rough metal surface that sometimes hurts when I'm lifting pretty heavy.
If you end up getting a pair for push-ups, you may want to get some that have more rubbery-sticky palms if you are doing push-ups on a smooth mat or floor type surface. As an added bonus, they probably give you some extra grip too.
I just bought the cheapest pair at Dick's, which have a soft fabric palm, and they are fine for push-ups on the carpet and for minimizing calluses in general, but I don't like wearing them at all for push-ups on mats or tile floors because it feels like my hands are about to go sliding sometimes.