I sympathize. My solution is a bit "much" but if you're looking for things to try:

I wear my gym shirt while bicycling which I sweat into profusely. I get to work and lock myself in the family bathroom, where I take my gym shirt off. Then, I take out a linen towel I bought that packs super tiny, get it wet, and wipe down. I stand and cool and dry off for a bit, then put on the shirt I packed. The linen towel gets a soapy wash in the sink and then wronged out. Then, towel and gym shirt get hanged on hooks in the office near a window (as far as I can tell there's no bad smell and I have explicitly asked others in the office) and both are dry within a couple hours.

Change into gym shirt after work and bike home, take a shower.

It's a process but it's worth it, I've lost 7 pounds without much else lifestyle change.

The linen towel is fairly critical - packs small, light, dries quick and is odorless (allegedly linen is anti bacterial). They're also great for travel in general.

Great ideas. I also have started bringing in an extra pair of shoes - loafers. I sweat into my hiking shoes that I wear into work, take them and the socks off so they can dry, then wear loafers around the office. At the end of the day I reverse the process.

Really depends on the climate - I commuted to my last job via bike, and it never would have worked in the summer without a shower at the job.

Get an ebike

That avoids much of the exercise (but has its own benefits).

I've heard it makes you more likely to actually bike around since it is easier.

Exactly. eBiking > not biking

From what I've been able to gather, on average you exercise about a third as hard. If it makes you bike three times more often, it's worth it. It also means you will be bothered significantly less by wind or other variability in travel time (maybe you are tired that day, for example).

Even a third might be being generous. I have never once really broken a sweat riding an eBike even in hot summer weather. I've actually had to put jackets/jumpers on to stop the wind chill on hot days - this never happens on normal bikes due to the effort exerted.

This however is precisely why I have some ebikes in addition to human steam powered ones - its so nice for commute or journeys when you want to arrive feeling refreshed but not need a shower or change of clothes.

Avoiding much of the sweat of exercise would be the point, no?

A lot of people pretend it is the tour de france. I just ride slower and take my time.

Something is triggered by there being a bike just in front or (much worse) just behind. The more dressed up and the better the opposing bike (somehow they are in opposition) the greater the effect.

I would still sweat. I sweat just getting out of the shower or standing in a warm kitchen too long with the oven on.