I started a new game on story mode and progressed to the point where get control of your mech bay. (I was trying to do this with my Dire Wolves from endgame but I still needed 2-3k more xp. You might be able to do this at endgame if you go all in on a mech when it first becomes available so that you have 5 or 6 missions to get xp)
I did this with Arctic Cheetah Light Mechs. You start with two in your roster, so I bought 2 more with the starting money available. Stripped out all the weapons and ammo and installed 4 Medium Pulse Lasers on the Prime Loadout along with a .25 armor mod in each arm to max out tonnage. No additional heat sinks are needed, the heat is very manageable.
Now you have 4 Arctic Cheetahs and what ever other mech you want to use for a few missions. I was running the odd mech (Kit Fox), so I tried letting my teammates in the ACs get all the kills as much as possible to fill out those Milestones. I think I did this for 2 missions until I hit Honor 3, where I could buy my 5th Arctic Cheetah.
I used these 5 ACs for the next 3-4 missions, 2000 XP for the 5 mechs per mission and milestones completed for 1k each. By the time I hit Honor 5 I had enough honor to purchase all 30 upgrades (5 tiers of 6 stats). The only milestones I had left were Heavy and Assault Mech kills; 7 milestones completed for 7k xp.
Happy Hunting