If you prefer something to soak up sweat, go with a headband. The head band could be cheaper and of course, less material means less of your head is covered to cause rise in heat. I'm sure Decathlon will have something cheap and not make your head explode from heat.
If you do buy a hat, just make sure you look at the materials label! For example:
100% cotton is great for keeping your head warm and soaking up sweat, but a no no for insulation.
Try looking for a a material make up that's similar to your jersey. Materials such as nylon or polyester with some elastane. Only example I could find is Rapha. FYI do not spend anything close to this much for a hat if you plan on buying one.
IT COULD also be your helmet is not well insulated (doesn't have enough holes) resulting in your head heating up and thus...sweat.
Once you go bibs, you won't go back to shorts. In order for shorts to stay on you, it's snug around the hips, thus, your beer belly sticks out in any aero position. Whereas bibs uses the straps to keep them from slipping. Most bibs will cover that beer belly and for the better lack of words, so none of it is hanging out.
It will feel very very weird at first when you try it on because the last time you wore overalls you was probably those hideous jean overalls mom made you wear back in grade school. Knowing where you grew up in the states, you know I'm right.
You won't really feel the difference when it comes to them being too hot. Lots of the materials on the bibs from the waste up will be very well insulated.
Check out REMA who does have a retailer in Taipei (over by Jian Tan) and also having a 30% off sale. They may have your size! You can also order online and do exchanges if you don't like it or need to exchange sizes. I have a pair of their tier 1 (ㄧ級) bib shorts and they are awesome for the price tag.
You can also look into Atlas and Frontier for other options in the same price range.
I know you're tight on budget with the kids as well as saving for those new wheels, but make sure the bibs you're buying are at least 2000-2500NT range. Do not, I repeat, do not buy bibs from Decathlon. If they are your only bib and you'll be wearing them during your 40km riverside loops 2-3 days per week, they will rip in about 8-9 months.