We arrived at the border with expired transit visas (3 days validity). According to all officers at the Kobero border, we were initially supposed to prolong them in Bujumbura, where the immigration office was supposed to be open on weekends. We specially asked for that information on our arrival at the Kobero Border a few days ago. Well, it was not open even on Friday afternoon, so we missed our chance to extend it.
We have contacted our former fixer from the Kobero border, who said he knows how to exit legally. We picked him up in front of the border and drove with him through several groups of police officers, who probably wanted some money, but since he was sitting in a cabin, they just smiled and said something to him. Once we arrived at the border, we did a health check and found out, to our surprise, that he had not fixed anything with any immigration officer there. To make a long story short, whatever he promised was not true, and it was another scam that cost us 40 USD and 46000 BIF.
In the end, we got help from the Muyinga governor and got the exit stamps. Then we got stamped our CPD and moved to another window to do a Tanzanian transit visa (no forms, 30 USD fee paid in bank). After that, we stamped our CPD with Tanzanian Customs and left the crazy border (no TIC). The whole procedure took around an hour, and nobody searched the truck. Just at the exit gate, we opened the door for some Customs officers and left after a friendly chat.