Eh, I don't have much problems with this, but that's probably because I use some of the means provided in-game. The catwalks are a godsend when trying to mount a mech in low-gravity, and I'm missing them sorely in Hel's Gate.
So, when inside the main base, make a guess as to which bay your purchase is going to spawn in, and then run around the catwalk behind its back. You'll notice there's an opening on the catwalk you can just run off to get the flat of the mech's back, which you can do fairly reliably without getting yourself hurt to any significant degree.
Hawkmoths have the flat part of the fuselage near their nose, so just ram into that 'til you get the prompt to enter.
Aerospace? I don't have much experience there, but try going for the mid-body of the Sulla and the nose of the Shiva. The idea is to avoid sharp edges, just like in real life.
And remember, it's not your fault. Battle armor is just ridiculously delicate.