I would leave Manasha at the vet for an additional day to wait for the thermostat. Pet store thermostats can be 'iffy'. So I would pass on them.
Heat mats will not change ambient temperature, but your 77 ambient temp sounds fine. I would only worry if the temp drops below 75.
And an unregulated heat mat will skyrocket to dangerous temps, so thank goodness for thermostats!
Hope that Herpstat arrives soon.
Please don't put the probe in the bin. You risk damaging the probe with tub moisture, feces, and urine. Not to mention that any of these factors can mess the probe reading. Also the animal has the ability to move the probe as well, subjecting a temp change, which can be dangerous.