Narita Express will be more comfortable to NRT from Shinagawa, especially with large luggage. One seat and they come around with a snack trolley you can buy a snack to eat onboard (there should be drink vending machines aboard). If you're going to stress about it, that's the better option. But it's a long journey, >1hr.

NRT is less crowded overall and may have shorter security lines. I've been to the ANA lounge and was not very impressed relative to the JL lounge which is my usual. The United Club is ok in terms of seating and quiet.

HND is a very short train ride from Shinagawa on the Keikyu line but it's a regular commuter train. I did it 2pm on a weekday recently and it was not too crowded. I can't comment too much on the *A lounge situation at HND. Maybe try both airports coming in and going out and see how you like them. But otherwise I'd pick the more convenient flight option (plus one that maximizes time in Tokyo if that's a priority) and not stress too much about the airport transfer.

I'm not a big fan of the Friendly Limousine Bus because they can be slower than the train and can get crowded but Shinagawa -> HND is not that far.