Lyn M. Schlitt, a Washington trade lawyer who lost out in a battle between Congress and the White House after President Reagan nominated her to a seat on the International Trade Commission, has emerged as the leading candidate to become general counsel of that quasi-judicial body.

She passed muster in sessions before the five commissioners, who submitted her name as their choice to the Office of Personnel Management, which will check her qualifications for the job, according to sources among trade lawyers.

Schlitt, a political independent, is a trade specialist with the Washington law firm of Covington & Burling. She has represented both importers and domestic industries in cases before the ITC.

She was considered the most qualified of three nominees that President Reagan submitted to the Senate in early 1983 for seats on the ITC. But she ran into opposition from labor interests because she had represented Honda Motor Co. before the commission.

There was more to it than that, though. The White House was most anxious to win Senate confirmation for the other two nominees, Susan Liebler and Seeley G. Lodwick and traded away the Schlitt nomination to give Democrats on the Senate Finance Committee a bigger role in filling the third vacancy, which had to go to someone who was not a Republican.

If OPM raises no objections, Schlitt will replace Michael H. Stein as general counsel. HOUSE-HUNTING . . .

The ITC is still searching for a new home now that Congress has turned over the historic Old Tariff Building to the Smithsonian Institution for use as a gallery for art shows and other exhibitions. The building at 7th and E streets NW has housed the government's trade agencies for 63 years. But since 1968, the Smithsonian has coveted the building to augment its programs at other galleries in the neighborhood.

The General Services Administration had planned to relocate the ITC to a building at 500 North Capitol Street, but it was unable to reach a leasing agreement with the owners. BIG NEWS IN CANADA . . .

Most Americans may not know much about the ITC, but Canadians are becoming increasingly aware of it as trade frictions intensify between the United States and its northern neighbor.

The Canadian Broadcasting Corp. is doing a special television program on the ITC, sparked partially by two recent fact-finding investigations into imports of fish and pork from Canada into the United States. Both those reports have been in heavy demand.