I had DD at the end of May last year and she spent the best part of the day in the HAB during the first 3 months or so. And that is in South East Queensland... and Maya was a hot and sweaty baby. I usually dressed her in either a singlet and a All-in-One (Just a cotton one, not a towelling or fleecy one), or, on cold days I put a long-sleeved bodysuit under the All-In-One. I also dressed myself a little lighter. She often had warm, red cheeks, but she didn't usually get upset. She absolutely adored the HAB. I guess they do get quite warm cause most of their body is heated by your body heat (while only a small part of your body is up against hers, IYKWIM). If however she even gets upset when she only wears a short sleeved body suit, then maybe she's getting upset about something else? The act of getting upset makes most babies hot and red in itself, so maybe she's hot because she's upset rather than being upset because she's hot???
I wouldnt worry too much about overheating her, to be honest... Just follow her clues. If she gets upset for whatever reason, take her out of the HAB. DD loved the HAB, but once she got a little older, she didn't like being too restricted while awake. She loved being really snug when sleeping in it, though.

You can also try and go for a walk outside to cool her down???

Sasa