Of the ~35 teams that are not auto-bids, typically only a handful have losing conference records. But think 9-11 or 8-10. 4 games under .500, maybe one or 2 a year TOPS. It would totally depend on the bubble this year, how deep it is or isn't.
I would just say, it's possible, anything but a lock.
Historically speaking - the Big12 is nearly an annual 7-bid league. And, almost annually a team with a losing record is continuing to the big dance. In recent history, of the 7-11 teams one has made it and one has not.
2020-21 - 7 Big12 teams made tournament, all with winning conference records (9W-8L or better thanks to ISU going 0-18)
2019-20 - no NCAA tournament
2018-19 - 7 Big12 teams made tournament, one was 7-11 in the league and made 2nd round. The other 7-11 team (8th in Big 12) made NIT Final 4
2017-18 - 7 Big12 teams made tournament, 6th-9th tied at 8W-10L. Two made big dance and ended the year 19-15 and 18-14. The other two made NIT
2016-17 - 6 Big12 teams made tournament, 7th&8th were 6W-12L, including TCU who won NIT after ending conference play on a 7-game losing streak
2015-16 - 7 Big12 teams made tournament
It all depends on the top of the conference. If the top 2 teams finish something like 16-2, then 7-11 could finish 7th or even 6th in the league. The opposite happened a year ago, ISU finished 0-18 and 7 teams finished the year with winning records.
If we pair 13-0 non-conf with a 7-11 Big12 season, and a couple of those 7 being the ranked Big12 teams at Hilton, it will be enough. If we sweep the bottom 3 teams to get 6 wins, and only get one win from the top 6, then it isn't enough.