How to register Russian tourist visa while on the Trans-Mongolian railway?

My partner and I will be starting our Trans-Mongolian adventure in about a week or so, but after reading the visa registration procedures from various resources, we're still a little confused about some of the details for our trip.

Basically, we'll be travelling from east to west. More specifically: our journey starts in Beijing, after which we'll hit Ulan-Bator in Mongolia for a few days, and our first stop in Russia will be Ulan Ude. From there, we'll be making our way towards Moscow over the period of 2-2.5 weeks.

Our main uncertainty is how to deal with registering our tourist visa along the way, from Ulan-Ude to Moscow (once we're in Moscow, we'll be staying at the same accommodation for a week, and close to another week after that in St. Petersburg, so we're confident our ho(s)tels will be able to help us out there).

Between Ulan-Ude and Moscow, we'll only be staying for one or two nights at a place at a time. That is, we'll get off the train, stay for one or two nights, and continue onwards west by train after that.

Our two main questions:

• Is it fair to assume all hotels and hostels in Ulan-Ude offer a visa registration service? As far as we understand the process, we'll definitely have to register our visa within 7 days (not counting weekends and public holidays) of entering the country, regardless of how long we're staying at one location. Ulan-Ude seems to be our best option/most logical choice, but if not, what alternatives do we have? We were informed that sometimes the local UFMS or post office requires you to present a formal letter of invitation to register a visa - we don't have one, and honestly, it's sounds like a painful and time-consuming process if we were to have to do this at every stop...

• After Ulan-Ude, can we expect visa registration services from ho(s)tels in cities like Krosnoyarsk, Novosibirsk, Perm and Vladimir? Should we even bother registering if we're only staying for one or two nights at every stop, or could we cherry-pick one or two cities where registering is 'easy' (while making sure we don't exceed the 7-day period between every subsequent registration)?

We're sure we're in a pretty common situation, and we're probably worrying too much about this stuff, but for whatever reason we found little to no documentation on the appropriate course of action for this scenario. Any advice is greatly appreciated!

PS: We're doing this trip independently. There's no travel agency or so involved, but of course we did get our support documents through a licensed tourist agency based in Russia (based on which we were both granted a 30-day tourist visa).