I would say it's a very slight increase in speed. On large maps princess still takes a very long time in the move phase. I pulled a super-nerd maneuvor and played Aurora at the same time as MekHQ while I waited on Princess's moves.
One thing you could try is picking a small map in the lobby; small maps reduce the calculations Princess does in move phase.
Another thing I've done for some large battles is have the player forces show up late as reinforcements. If you do that you click the end phase button while the AI does it's thing for 8 rounds or so until you show up and have to click more stuff.
If you set the player forces as bot controlled you don't have to even click phase end buttons.
Not ideal I know but that's how I played the story campaign I'm playing. I started as an ultra-green teenage mech inheritor, made him Founder, and then stuck his Stinger in a training lance with a veteran level lieutenant taking cadre duty contracts until he was around 18 and I made the actual unit. The only missions the training lance was getting were Special Missions because I set the intensity on training to 1% to reflect a rear area garrison contract in exchange for pilot training for independent mech owners. You can set the contract payout low so the trainee unit is earning maybe 40-50k C-bills a year which makes sense as he's mostly benefiting from the training. I do a last contract with more fighting as a sort of graduation contract before I start the actual unit from the trainees.
The officer found a surprising amount of Star League caches. I played him like Sgt Unther from Mechwarrior 2; going out on lone wolf patrols for bounties for that one bed, one bath back on Galatea. He started with a nonsensical weapon specialization (a primitive version of something) so I changed it to Large Laser because he started in a Phoenix Hawk 1K. He was shot out of it on one of the Special Mission "patrols" (since my other pilots were all green). Since he managed to get a few kills, I RP'd that the merc unit the training lance was attached to gave him a new ride and rolled up a rare Sentinel for him. After being shot out of that one, I bought him a Phoenix Hawk legit off the unit market which he died in, leaving behind a wife (who was in the personnel market, owned a mech, and was green so I recruited her) and three children. The wife joined the new merc unit because I actually liked Liuetenant Xuan, the deceased officer, callsign Coach.
I found a Phoenix Hawk LAM early on, can't remember if it was salvage or on the unit market. The Founder is driving it around right now and I'm putting points into Aerospace stuff. He's still ultra-green in flight mode, which is probably ok because I haven't figured out how LAMs work yet.
EDIT: Currently year 3026. I think I started around 3020. I chose that time so the Founder would be middle aged and with a gang of adult children in various roles when the Clans show up. I tried doing that even earlier in 2980ish but it took forever once the unit bloated up to regiment size so I started in the early Renaissance this time.