Well first, don't worry about monitoring the temp strip constantly, it's not crucial to be dead on a certain temp in a swamp cooler, just within a low range.
You also really only need to maintain the temp for the first 3 days of active fermentation (if there is a 72 hour lag time before that that call it a week to be safe) after that temp control is not crucial.
Secondly if you've kept your fermenter sealed and the co2 bed intact moving it around won't harm it within reason. You just don't want to get it sloshing too much that you incorporate oxygen and will hurt your beer. You're beer's not that week.
But really just at this point, just relax and let the yeast do it's job, you don't need to hover.