Commuting in "Work Clothes"?

It should be fine if you take it easy. If you don't want to take it quite so easy just take a change of top and some deodorant. If you wait 5/10 minutes after you arrive to change you should have cooled down sufficiently.

Alternatively, build up to cycling the whole route and embrace the lycra.

get an e-bike?

also, steep climbs on the way back isn't a big deal is it?
you can just roll down them on your way in.
there shouldn't be any sweat generated by braking...

I do a bit of cycling in work clothes; normally when I'm in smart/casual mode, although I have ridden in a suit. Often it's when I'm away from my normal place of work and haven't got the usual cycle facilities. Keeping the speed down is the key, which isn't that easy to do when you want to go with the traffic flow (safer). The other thing I do is to try and start the ride feeling slightly cold (depends on the weather) and leave off any jackets, jumpers etc, assuming I'll warm up during the ride. Allowing a few minutes to cool off outside is also a good move which saves walking inside and breaking into a sweat. More technical fibres work better than cotton as well. I have a pair of smart(ish) Rohan trousers that got soaked from the thigh down in a cloudburst. 10 min walking around indoors and they were as dry as a bone. When I cycled in a suit I thought it'd be a disaster but it turned out to be no problem at all. It's not hard to look smart where I work given how scruffy some people look, and they drive!

I do a bit of cycling in work clothes; normally when I'm in smart/casual mode, although I have ridden in a suit. Often it's when I'm away from my normal place of work and haven't got the usual cycle facilities. Keeping the speed down is the key, which isn't that easy to do when you want to go with the traffic flow (safer). The other thing I do is to try and start the ride feeling slightly cold (depends on the weather) and leave off any jackets, jumpers etc, assuming I'll warm up during the ride. Allowing a few minutes to cool off outside is also a good move which saves walking inside and breaking into a sweat. More technical fibres work better than cotton as well. I have a pair of smart(ish) Rohan trousers that got soaked from the thigh down in a cloudburst. 10 min walking around indoors and they were as dry as a bone. When I cycled in a suit I thought it'd be a disaster but it turned out to be no problem at all. It's not hard to look smart where I work given how scruffy some people look, and they drive!

For 20 minutes cycling you're going to spend more time getting changed etc. than on the bike. So it doesn't make a lot of sense to wear cycling gear.

Either cycle more of the journey (all of it?) or cycle the shorter version in work clothes, gently.

I cycle in occasionally, to get the miles done in an effort to get to 100 miles/week, just under an hour each way (not going gently). The hassle with taking in clothes, shoes, working out if I can leave my laptop etc. etc. etc. is only just worth it.

I commute 16 each way 4 days a week, inconvenience of showering at work? Wrong, it means you can fall out of bed into cycling kit, ride and eat, and by the time you get to work you are about conscience enough to risk having a shower.

Curious as to why only 4miles? I would almost say it is hardly worth it? Unless you are intending to gently build the distance, please don't take this as dismissive, but unless the drive is a long one, and I do know someone who used to drive 15miles then ride the remaining 15, 30 being a bit too much! 4 miles doesn't really sound like much if you are using a car as well?

Cycle all of the journey then?
Means you dont sit in the car in cycling kit then.

If you want to commute in work clothes and not sweat then I can suggest perhaps cycling more slowly? Don't treat it like it's a race.