I feel like I need to clear up some possible misconceptions about the use of the terms 'Bluetooth GPS' and 'GPS Loggers'.
They are two very different things and a device does not nessasarily do both.
I know the OP was looking for a BT GPS to use with GPS-Logit for posting to the GPSTC.
Unfortunately, as far as I can find, there are no currently made commercially available GPS BT devices that are suitable for the GPSTC.
There used to be at least one, but it was discontinued by the manufacturer around 10 years ago:
Wintec G-Rays 1:

This device is not a Logger! It does not have any way to record the data, only to send it to another BT enabled device like GPS-Logit Android Phone App. It used the Ublox 6 chipset and can be set up to output Doppler Error data (UBX sentences) and can be approved for GPSTC posting.
Here are examples of custom home made devices that do exactly the same thing but are based on the later M8 multi GNSS ublox chipset that
can do Muliti GNSS data up to 10Hz and GPS only data up to 18Hz (The same chipset the Motion uses).
Made by Manfred Fuchs:

For Scale (the G-Rays 1 was even smaller):
Made by Mathew Robertson:
If anyone knows a good electronics fabricator or hobbyist, these devices are not that hard to make. There is essentially 3 components. The M8 GNSS module, the BT Module and a battery with some sort of charging connection - oh, and a switch for on and off.
To make something into a Logger, one has to add a few other things that make it a bit more complicated. Decrepit and Boardsurf can fill you in on that, but here is a pic of a very neat and compact Twin GPS, 10 Hz logger that Manfred made and I tested for him against the timing gates at Luderitz in 2013. This one was based on two Ublox 6 modules and each had a seperate SD card to record the data.