If Bayushi Bob, a Bayushi Deathdealer marries into the Kitsuki family, for example, can he continue his school training if he buys the Support of Bayushi Family Distinction?

Wouldn't be an unreasonable way to handle it, imo, yeah.

You're not forced to do that.

Rules-wise, you can still continue your school training regardless. You pick a school, and stick with it - if you have an out-of-family or out-of-clan school you should have a narrative reason why - and your choice of advantages is a perfect way to illustrate it.

Deciding whether to create that character as Scorpion-Bayushi-Deathdealer, or Dragon-Kitsuki-Investigator, or some combination of the two is a good way to mechanically illustrate how long the character has been living with their new family and to what extent they've 'gone native'.

A former scorpion who's maintained close relations with their old family would be perfectly justified in taking "Support of".