I don't know too well TSC3, but it seems quite similar to other WinMobile field controllers.
The Bluetooth transceiver is 2.0+EDR compliant, suitable with the most part of GNSS new and old receivers.
So, You can connect the i-80 with TSC3.
I checked it and SurvCE run on TSC3. Furthermore, this software is suggested by iGage and therefore it will surely be compatible with the receiver i-80.
Sorry about late response.
If you haven't found the right answer yet, I think this might be useful.
CHC i80 works with SurveCE Version5 & CHC Landstar 7.
TSC3 works with SurvCE V5.
You'd find this info on http://www.survce.com/supportedHardware
Not sure if TSC supports CHC Landstar7. But SurvCE V5 is a better option.
So, you can use TSC3 for CHC i80 with SurvCE V5.
If you're using TSC3 with Trimble Access, now Access fully supports Spectra Precision SP80 as well.