Yes, and at BKK you need to follow the signs for the transit immigration to domestic flights, don't go to the regular immigration queues for people heading into Bangkok. No need for fast-tracking doing it this way. Normally TG international incoming flights show the arrival procedures on the cabin screens as you approach BKK, so pay attention. Once in the terminal, things are well-signed. Assuming bags are tagged at FUK all the way through to HKT, bags will be auto-transferred so you don't deal with them and Customs until HKT.

Reverse is even easier if just connecting in BKK to intl. Immigration and Customs dealt with at HKT, as should boarding passes. Then nothing more to do once you get to BKK except enjoy airside (and the TG lounges if on intl outbound).

No need for a third-party service at either BKK or HKT.