I am using a company called nava-rent.es. email email@example.com. Five day one way bike hire from Logrono to Leon is costing €150 per person including panniers/helmet/repair kit etc. I have been in touch with the guy (Javier) he's seems decent and will give you a quote by return of email. Look at Gronze.com and check out the elevations and distances to give yourself an idea of what's involved. Three days from Burgos to Leon should be pretty manageable.
Peg keeps mentioning that if she did the meseta again she would bike it. I am blond and blue-eyed and for decades I've said that I like sunny days because you can find out where the shade is easier. However I really liked walking the meseta. I wore light nylon long-sleeved shirts and pants and a billed cap. Since I was walking west in the morning I wasn't looking into the sun.
Often I tell people to look into renting a bicycle (mainly because they are pressed for time) but here I'm suggesting that the walk may be okay.