I think a Wesley character could work - have the whole ambiguity about if he is or is not Picard's son, leave it open since Beverly is not on the Ent-D. However, Wesley is a B-character appearing to cause Picard uncomfortable moments, get video message from his dead 'Dad', and helm the ship since he is an cadet in his final year.
Wesley can be on away teams and stuff to make up the numbers and occasionally say stuff, maybe even save the ship once in S1. However, he is not alone and there is a whole 'class' of cadets doing their final year 'in the wilds' - this is so Wesley has someone to talk to, also allows for Lower Decks (episode not series) style stories.
I would perhaps have Wesley injured near the end of the season and have the truth of his parents come out. I am leaning towards him being Picard's son, but am not 100% sure on that without Beverly there for the payoff. Perhaps save the mystery for later in the show if Bev joins?
If the fans take to Cadet Crusher perhaps he comes back in S2 as Ensign Crusher, specialising in command track?