It should be possible to do everything on the airside of Changi Airport. SQ operates from Terminals 2 and 3 while I think TG flies from Terminal 1. There is an airside Airtrain that connects all three terminals.

Just to be clear, are these two sectors on the same ticket or are they on two separate tickets? If it's all on one ticket Ideally what you want to do is to have SQ i(at check in at HKG) issue all the boarding passes including the SIN/BKK sector on TG. As part of this process any check in baggage should show final destination as Bangkok. If this is possible then you have no issues. Upon arrival at Changi you just need to get yourself to the gate for your TG flight. This is all on airside.

If it's two separate tickets or SQ cannot issue the SIN/BKK boarding pass at least try to get them to tag your luggage to BKK. In the two separate ticket scenario, you will need to show them your SIN/BKK ticket so they can make proper reference to it in their own computer record.

If SQ can only issue the boarding pass and the baggage claim to Singapore, you might want to go to the TG check in counter or alternatively go to the TG Royal Silk Lounge if you have access (both at HKG) to see what they advise. Otherwise, on arrival at SIN, go to the transfer desk on airside, look for the counter for TG. They will be able to deal with all this.