The House Democratic leadership was badly beaten on one major issue yesterday - college tuition tax credits - and remained in trouble on another: its controversial campaign financing bill.

The leadership tried to keep a tuition tax credit amendment off the floor and instead force through an alternative that President Carter favors, which would simply increase existing federal scholarships and college student loans.

The leaders needed a two-thirds vote to bull through their bill. They failed even to win a majority, losing 218 to 156, with Democrats as well as Republicans voting against them.

They next were forced to cancel a Rules Committee meeting scheduled for today on student aid legislation. They had wanted the normally docile 16-member committee to forbid a tax credit amendment on the floor, but all five Rules Republicans were balking, and so were four committee Democrats.

So the college aid question has been put over until Congress returns next month from its Easter recess.

The Rules Committee did report out the campaign financing bill, but whether the rule the committee imposed on floor consideration can be adopted remains in doubt. To kill the rule is tantamount to killing the bill.

The bill would make several important but rather technical changes in existing campaign financing laws. Some members also intend to propose on the floor an amendment providing for public financing of House elections.

House Administration Committee Democrats earlier this month approved controversial amendments that would prevent Republicans from making full use of their advantage so far this year in raising campaign funds.

Those amendments raised such a storm that the Democrats, led by Administration Chairman Frank Thompson (N.J.) have promised to drop them. But Republicans now think they may be able to kill the bill altogether, which they would prefer.

The rules committee was scheduled to meet again today on the student sid bill, but after Democratic members lunched yesterday with the Speaker, the meeting for today was put off, and the issue is not likely to come up again until after the Easter recess, which begins Thursday.

Meanwhile, the Rules committee did report out the controversial campaign financing bill, but whether the rule the committee reported can be adopted remains in doubt. Killing the rule is tantamount to killing the bill.

In a "Dear Colleague" letter, Rep. Guy Vander Jagt (R-Mich.), chairman of the National Republican Congressional Committee, said, "Let me assure you that, in spite of Mr. Thompson's latest sleight-of-hand manuever, little has changed, certainly not enough to motivate House Republicans to drop our unified opposition to this latest power grab."

Minority Leader John Rhodes (Ariz.) asked the Rules Committee to return the bill to the Administration Committee, citing "other aspects of the bill" he wanted reviewed.

Meanwhile, 106 Democrats signed a letter generated by Illinois Rep. Abner Mikva, chairman of the Democratic Study Group, attempting to promise the GOP that they will vote for Thompson's amendment to raise from $15,000 back to the present $50,000 the amount a party committee can spend on a House race.

Many DSG members support a pending amendment that would provide partial public financing of House races, and killing the bill would prevent a vote on that.

In both issues, Democratic tactics appear to have given the Republicans - who control only one third of the House - the upper hand, at least tempararily.

The controversy over the student aid bill involves a race between two competing proposals - a tuition tax credit sponsored by Republicans, and a rival Carter administration plan that would increase federal college scholarship aid instead.

The tax credit plan, which would aid parents of elementary and secondary school pupils as well as college students, has gained support rapidly in recent months as a way to help povide tax relief for middle-income Americans and those attending parochial schools.

Carter unveiled his own plan early this year as an attempt to head off enactment of the tax credit legislation, which the administration contends would be too costly. The tax measure would cost $4.5 billion a year, compared to $1.46 billion for the Carter proposal.

The tax credit approach has been endorsed by the influential Catholic school lobby, which would not benefit from Carter's proposal. However, public school officials - and the Democratic congressional leadership - are pushing for the Carter plan.

On Saturday, Attorney General Griffin B. Bell issued an opinion contending that the tuition tax credit would be unconstitutional because providing aid to parents of parochial school students would violate the principle of separation of church and state.

However, Bishop Thomas C. Kelly, general secretary of the U.S. Catholic Conference, took issue yesterday with Bell's assertions, and called for enactment of the tax credit legislation. The tax bill is sponsored by Sens. Bob Packwood (R-Ore.), Danield P. Moynihan (D-N.Y.) and William V. Roth (R-Del.).

The tax-writing committees and education panels in both House and Senate are racing to get their own rival bills to the floor ahead of their competitors, on the theory that whichever piece of legislation makes it to the floor first has the best chance of passage.

Both bills now are ready for floor action in the Senate. However, in the House, only the Carter plan has been cleared by committee. Proposing the tax credit bill as an amendment to the Carter measure would have been the only way to force consideration of both at once.