Has a majority of one political party in one house of Congress ever voted in opposition to the same party in the other house in the same Congress?
Yeah, while certifying the presidential election in Jan 2021:
A hundred and thirty-nine House Republicans and eight senators voted against certifying some of the Electoral College votes [...] on January 6th.
139 is a [slight] majority of the 213 [or so] seats that Republicans had in the House. I'm not exactly sure why, but Reuters puts that latter total at 221, which still makes 139 an intra-party majority in the House. (The Reuters total of 221 doesn't even seem to match the pictures they show, which are only 211, by my count. So 221 is probably a typo.) There were 51 Republican Senators at the time; this went down to 50 only on Jan 20. So 8 Republican Senators was [rather clearly] an intra-party minority. The votes were actually taken by state (election). The numbers reported in the press were probably an aggregation of the two states results that had objections registered. More precisely
Republicans objected to the AZ election results: 121-83 (House) and 6-45 (Senate); for the PA election result: 138-64 (House) and 7-44 (Senate). Furthermore, initially, about 14 Republican Senators had informally announced they'd object to at least one state result (they still would have been a minority among Senate Republicans, barring some contagion during the actual vote). But this number apparently went down after the invasion of Congress by Trump supporters.
But note that this is technically not "on some bill", as the rest of the Q asks. (Also, the certification happens as a joint session.)
Inspired by CDJB's answer, I've looked at the other debt ceiling extension bills. And sure enough, in 2002 most (36, if I counted right) Democratic Senators voted for it (14 opposed though), whereas in the House the Democratic vote was 3-206 against. Interestingly enough, the next year's  extension was however overwhelmingly opposed by Democratic Senators as well--only two voted for it. But it passed the House via a "deem and pass", i.e. with no explicit vote on it.
The 1984 ceiling increase also has a somewhat hilarious story behind it. The Congress record shows it as:
10/12/1984 Passed/agreed to in Senate: Passed Senate without amendment by Yea-Nay Vote. 37-30. Record Vote No: 292.
10/12/1984 Failed of passage/not agreed to in Senate: Failed of passage in Senate by Yea-Nay Vote. 14-46. Record Vote No: 290.
10/01/1984 Passed/agreed to in House: Passed House by Unanimous Consent.
According to the contemporary NYT article, in the first Senate vote of the day, the Democratic Senators had decided to teach Republicans a lesson, because they had been campaigning on the trail against the limit increase. So they forced Republican Senators back to Congress for a vote, which they had not expected because of the unanimous passage in the House (as well Democratic Senators support in the prior years).