I'm in PME and I have acquired the troops from FE and AF so far. I found a modified approach here for AF. I was able to use only 4 research techs and still got the maximum troops: 14 Recon Raider, 11 Surrogate Soldier, and 10 Battle Fortress. I used a mix of fighting and negotiating to do this.
The key factor is in reading through the story quests as documented in the Wiki, the claim was that for many of the quests that are different for fighting vs. negotiating, that only what you did on the most recent province(s) mattered, so it is possible to take a "mixed" route.
After going through the story quests on the Wiki, for avoiding the extra research tech quest, it wasn't clear for certain which quest caused this, but I strongly suspect it is the quests called "Getting Ready" which there are 3 and the latter 2 of them require to research a tech. (This is in stage 1 of AF map around when you scout Tuklavik.) My suspicion is that the difference is that if you negotiate, you skip the 3rd "Getting Ready" quest and avoid the extra tech research. And specifically, I believe you need to negotiate all the stage 1 provinces that are owned by the "Natives" to avoid the extra research.
The only provinces that it appears you *must* fight to get extra troops are:
Nilak (stage 3) -- 2 Recon Raider (quest name is "Cryptic" instead of "Origins" if you negotiate)
Nunataq (stage 4) -- 3 Surrogate Soldier (quest name is "Shallow Pockets" either way, but if you negotiate the reward is not troops)
I don't think there is a difference to account for 8 Battle Fortress for negotiating vs. 10 for fighting. In the chart for the negotiating route, the numbers still add up to 10, but the totals only list 8, so I think the negotiating total listed for Battle Fortress might be wrong here.
Someone may want to test all this more precisely, but here's what I actually did to only require 4 research techs and get the maximum troops.
Negotiated everything before Nilak
Nilak - Fought
Chulyin - Negotiated
Tattilgat, Atka, Nunataq - Fought all sectors
Others in stage 4 - fought some sectors for each province, but negotiated many sectors as well