Ditch the big clothes. Thin layers are better for temp regulation than effectively a duvet or a plastic bag. A Soft-shell jacket is far far more breathable and comfy to wear and still has enough water and wind resistance for 90+% of the weather you'll ever meet.
If you don't want to look bikey at all then consider Merino wool, t-shirt or jumper wicks sweat and doesn't smell and works well as an underlayer. Can.be costly but you can get good VFM effective merino in
Decathlon and general outdoor/hiking shops.
Go for rack and panniers or a saddlebag over a backpack, at the very least consider a backpack that sits off your back to allow airflow. Or maybe a satchel type over one shoulder and sits to your side bike bag/musette.
Entirely up to your own preference but if you have problems of head sweats & presentability after riding in a helmet then maybe consider not using it for a ride or two and see how you feel without.
A headband, buff or cycling cap can soak up forehead sweat, are less insulating and easier washed if you do feel the need for something on your head.
Consider how long you want to spend on the bike and maybe take longer at a slower pace or vary your route to get the most even and gentle gradients to minimise peaks and troughs in exertion and spikes in heart rate / body temp.
Some people, me including, just naturally run warm, it may be that you have to minimise sweatiness on the bike but still have to have a clean up routine for at work, a shower is ideal but wet wipes and a decent travelling miceofibre towel with deodorant will do for a sink wash and keeping your colleagues sweet.