I think it's quite a sad state of affairs that management jump to the conclusion that people will cheat. It suggests so many things and none of them are positive.
People are motivated by different things. For some, a bike ride might be a quicker way to work than a car or public transport. For others it might be the fresh air and a break from stress - but then again if it's in a busy city...... It could be fitness or it could be a money saver. Strava might appeal to the competitive ones and be a total turn off for others.
Then there's the fact that there are cyclists, people who ride bikes, people who never ride bikes and people who would never ride a bike. With 20 employees it can't be that hard to have a chat and find out what puts people off? And then work to overcome those. Of course, it would be helpful if management have a budget in mind.
Unfortunately I can see a whole lot of legal problems if work supplies bikes or carries out repairs or servicing.
I'd imagine (totally non-scientific) that fear or anxiety plays a part. Fear of an accident, fear of discomfort, fear of breaking down, fear of being the last, sweaty cyclist to show up. Jim or Jemima might be perfectly happy to cycle to work but perhaps their spouse isn't.
You know, with only 20 employees it would be possible for you as the "experienced" commuter to help them plan decent routes from home to work and back again..... And even ride with them once or twice, legalities permitting.
There's some good suggestions here - at a minimum a decent place to store bikes - but I think the best source are the people you want to motivate. Ask them.
And yourself. Arriving in fresh, chirpy and not feckin' and blinding about the traffic on the roads would help.
And Google. See what else has worked.
Good luck and please report back on progress if any.