Can a team with a losing record go to a bowl?
In short: Yes. The NCAA sanctions the bowls, so if they approval a bowl game, that game is going to happen whether or not there are enough teams with a winning record to fill the slot.
This year, UCLA is going to the renown Kraft Fight Hunger Bowl with a 6-7 record. Since the team has 6 wins, they’re bowl eligible despite a losing record (and no head coach).
When was the last time a team with a losing record played in a bowl?
North Texas (5-6), 2001, R&L Carriers New Orleans Bowl. North Texas won the conference, and thus the automatic entry.