Gavin,
I always thought that the strap on TPS units were aftermarket units? All the ones I have seen and the kind I have on my Mercedes and Hyundai are the type mounted with a rigid valve-stem and secured with nut at the base of the valve-stem. They typically have an O-ring to seal them to the rim.
If CopSta put on his (full size) spare tire on and did not get a low-tire indicator, that would indicate that he has a TPS unit in his spare. If he gets a low-tire warning, then he probably does not.
...Rich