Hi, I don't see any problems with you cycling from Santander to Burgos and you shouldn't have any problems finding accommodation along the way as most of the area is well catered for.
If you are constrained with time and want to take a bus to Burgos, I would suggest that you carry a few light-weight bike carriers for transporting your bikes. We have normal bikes (as can be seen in my photo, although I've since bought a slightly different one for myself), and if you take the wheels off they can go into a bag quite easily. Be aware that August can be horrendously hot, and even September as well. But that's one of the joys of being on a bike, isn't it as you make your own wind and can keep much cooler if you're cycling. And going over the Meseta is a doddle - so much easier than walking it. Carry lots of water and have some high-sustainable snacks and you'll be fine. Most of all, enjoy what is always a truly memorable experience.
We cycled from Bayonne to Santiago in July 2013, going through St John Pied de Port. Although it was hard in parts, we both found it a joy. Funny how the worst bits recede quite quickly yet you always remember all the great bits. And now we are looking into cycling from Santander using the Camino de Norte route and into Santiago sometime this year, and are doing a lot of research as we're both in our seventies now, and maybe not as fit as we used to be.