I can't even burn. When I manage to turn prograde (manually, the autopilot keeps turning the ship around and never reaches prograde or whatever) and try to burn, the ships slowly start to pitch down, although prograde button is on.
I tried both. The ship moves slowly and is very hard to control. With some practice I think I could control it, but the problem is I can't keep it straight when I burn, it always pitches down.
The built in autopilots only take your own vessel's weight distribution and thruster positions in account. If you change the weight distribution by docking a mass anywhere on the ship it will not be able to know what thrusters achieve what effect on your rotation.
Your docked mass is most likely not centered on your center of gravity. So when you burn your main engines the thrust is not centered and you induce rotation.
You can compare it to balancing a platter on a stick. Works very well, but if you put a weight on one side of the platter you need to compensate.
In planetarium mode you can see the actual forces involved.
Some (unmanned) sattelites/probes actualy rotate rather fast when using their main engine to compensate for an uneven weight distribution. This is rather hard to achieve manualy. Those vessels have carefully placed thrusters or SRBs to induce the rotation exactly along the desired axis.
I recommend you manualy control your attitude and gimbal your main engines to compensate for the uneven CoG during your TLI burn.