this swing effect is quite common. I personally don't like that much of a shift. There are a handful of solutions.
1 Proportional T-stat. The base models run about 105-135 $ it will resolve that problem.
2 heat the room up to 80ºF that will help slow the over shooting and slow response of the t-stat you have to the temps.
More thermal mass, glass works quite well it is slower to heat than plastic tubs and the temps inside will not raise as high as quickly and will take longer to cool down. You could use either the uth on a sheet of glass (polished edges in case it will prevent cuts) with the tub on that or just use a tank, the open tops have humidly issues that can be resolved by a cheap humidifier or misting. The tank can be easily turned on its side as well that will solve humidity issues as well.
Personally heating issues take priority (burns and failure to digest properly) over humidity (poor shedding)
I hope this helps some.
A