Smerel, rav aharon Kotler dealed, in fact, with mafia people to get money to save Jews in Europe. There’s a famous story mentioned in mishnas reb aharon with a mob boss who was so impressed with rav aharon that he didn’t even want his money back, and instead asked for a blessing. Rav aharon couldn’t wish him hatzlacha, because of his criminality, so he bentched him that he should die on his sickbed, and not from being killed. The boss was extremely happy with this blessing, and indeed survived many gunshots, dying at a ripe old age.
Was rav aharon being “mischaber” with the mafia? No. Neither sas rav michoel ber weissmandl when he worked with secularists.
The difference between the above and agudah’s shitah is that joining a government shows your approval thereof,not that agudah actually does approve of it, but they work on laws, committees, meetings, they are literally part of the state. Agudah holds this is ok for their reasons, and satmar says it’s not. That’s the machlokes. It has nothing to do with who you can work for and who you can make deals with to save Jews.