This is not directly supported in BLS.
BLS does support 1+1, 2+2 leagues where one-half of a league session is bowled against a different team and in your case, the team would be an individual bowler, or a team of one player. You could do the first two games of a session setting that up as a 1+1 (tell BLS that you are bowling two games per session, and then immediately select that you are bowling against two different opponents). But that would only work for games one and two...game three would have to be a separate "week" (can have the same date) but it'd have to be entered separately.
Given that you're going to do three opponents each night, my suggestion would therefore be to have each "week" (or session) a one-game week/session, and you'd have to enter just game one for all players on the first week, then advance to the subsequent week to enter everyone's game two, then advance another week to enter everyone's game three...but at least that way you could get all of the points to work out automatically.
Now, here's a rub. If you are using BLS-2016 (or successor BLS) PRO and wish to run Brackets, then you'll have to do the league as three-game weeks, manually overriding points, so that the Brackets module would have all three game scores in the same night. If you aren't doing Brackets, this won't be an issue.
I gave a relatively quick response without getting into the details of how you'd go into Setup:Rules to make this work. lf you already use BLS and can set that up, fine; if not, please say so and someone will respond with details.
Always check, download, and install the most recent update of your software to ensure that issues may have been corrected.