I agree that the Republicans would likely take a hands-off approach as well, but one major difference is that with a Republican in office, especially a noted tariff man like Harrison, using the tariff argument that McKinley employed in his bid for the Republican nomination in 1896 would fall flat. Even if the Democrats held a majority in Congress and blocked any tariff programs, Republicans would not be eager to set forth a nominee like McKinley whose entire career centered around issues of tariffs thus far. With tariffs discredited in the eyes of the party as a winning issue, the wing of the party obsessed with maintaining Gold Standard as their central plank, led by Governor Levi P. Morton of New York, likely take center field. If that were to occur, the Republicans are even more doomed in 1896 than they would be before, as the Gold Bugs would certainly lack the flexibility McKinley displayed IOTL on their main issues, and would be unwilling to compromise. Morton and whatever poor sap is chosen as his running-mate likely go down in ignominious defeat in 1896.
The real question, then, is which wing of the Democratic Party wins the 1896 nomination: the Bourbons or the Populists. The Bourbons were certainly not lacking in qualified candidates, with David B. Hill, William F. Vilas, Robert E. Pattison, and James E. Campbell counted among their numbers. The Populists, however, would have the up and coming momentum on their side, even if they lacked a head figure to rally around. Personally, I think the Bourbons narrowly win the nomination in 1896 without the OTL recriminations thrown against them due to better organization (although they probably give a sop to the Populists with the VP pick). Said ticket (maybe William Vilas/Richard Bland) goes on to quash Morton, but fail to end the depression. Come 1900, with Bland dead and thus the Populists iced out from a sinking Democratic administration, they decide to bail and run their own ticket (maybe William Bryan/Thomas Watson), which of course hands the race to whatever Republican is lucky enough to have decided to seek the nomination in the aftermath of Morton's disaster.