The hotel in SPB won't charge you for registering your visa, they are legally obliged to do so.
You will need to register your visa in Moscow if you're staying in an apartment, there is no legal way around that if you are staying more than 3 working days. You will also need some proof that you're actually staying at the apartment you say you are (letter from owner) I've added the address of the registration office at the bottom of this post, expect to spend at least 4 hours queuing and filling out forms.
One way around all of this is to check into a cheap hostel or hotel when you arrive in Moscow, dorm style accommodation can be had for about €15 a day, they will then register your visa for the duration of your booking, they will not care that you aren't actually staying at the hostel, in fact they may not even notice. However, this is not strictly legal but it's what I would do.
If you don't register your visa and you're caught you can be fined up to 5000 rubles, if you're caught a second time you'll probably be deported and will never obtain a visa again. Whether you get caught or not is down to luck, in years gone by I have stayed in Russia in private accommodation and didn't register my visa, I was never caught and passport control never said a word. They may be more strict now, I don't
know.
Good luck !
Address: #29, Leninsky Prospect, office 100 (first floor)
(Metro station "Leninsky Prospekt", corner of Leninsky prospekt and Stasova st.)
Phone: (095) 956-4422.
Open hours: Monday to Friday 9:30-6:00 p.m.
Registration price: only assistance with registration is provided (free), you'll have to register your visa by yourself