Any idea what the import taxes would be on a new watch, Tag Heuer in this case, and what the chances of getting it in under the radar?

Getting it in under the radar depends on how it's being brought in...mail. courier company, on your person, etc.  Import duty rate is 5%, VAT is 7% on the CIF value.

 

https://www.simplyduty.com/import-calculator/

best way to evade the radar is wear it on your person, but to be safe, don't have the empty box, receipts etc accompanying you on the same flight,
 

sending the empty boxes via courier also don't attract that much duty

Depends entirely on three factors:
1. The real market value of the watch in question and whether it's a first- or second-hand
2. The distributor you are ordering the watch from (thus the formalities he's ok to skip)
3. The ways he is able to ship the watch to Thailand (i.e. regular insured post, courier companies like DHL/FedEx, etc.)
I've imported quite a number of wristwatches to Thailand myself so should possibly be able to give some useful advice (PM me if posting in public isn't ok for you) 

Send it registered surface 'normal' (pay at point of departure postal surface/normal airmail with the insurance component added in case of theft.

Take off the watch band, remove the buckle from the band, mark it 'used watch and watch parts') this way you might avoid import duty.  

On 6/15/2023 at 4:11 AM, GaryPustynskiy said:

Depends entirely on three factors:
1. The real market value of the watch in question and whether it's a first- or second-hand

When it comes to import duty and VAT it is irrelevant whether the watch is new or used, the duties' calculation is based only on the CIF value of the item.

On 6/15/2023 at 6:44 AM, Tropposurfer said:

mark it 'used watch and watch parts') this way you might avoid import duty.  

Used watches attract import duty and VAT just the same as new ones do, the CIF value is what Customs are interested in.