Can the period be dropped in "vs" when used as part of a compound modifier?

I checked the two most influential (in the sense of "widely used by publishers") U.S. style guides— The Associated Press Stylebook (2002) and The Chicago Manual of Style (2003)—to see how they come out on this question.

The bad news for you (since you don't like the idea of including a period) is that both of them prefer including a period in abbreviations like vs. for versus. Here is the relevant entry in AP:

versus Spell it out in ordinary speech and writing: The proposal to revamp Medicare versus proposals to reform Medicare and Medicaid at the same time ... In short expressions, however, the abbreviation vs. is permitted: The issue of guns vs. butter has long been with us.

For court cases, use v: Marbury v. Madison.

And from Chicago:

15.4 Periods: general guidelines. To avoid unnecessary periods in abbreviations, Chicago recommends the following general guidelines: use periods with abbreviations that appear in lowercase letters; use no periods with abbreviations that appear in full capitals or small capitals, whether two letters or more.

The presence of a hyphen joining an abbreviated word that ends in a period to another word doesn't alter Chicago's advice at all. The only example of this type of construction it gives is "U.S.-Canadian relations," but the principle would remain the same regardless of how many hyphenated terms were strung together. AP likewise doesn't attempt to carve out an exception to cover situations where an abbreviation is included as part of a compound modifier.

The good news for you is that style guidelines don't deal in rules of grammar or syntax; they merely provide advice on consistent treatment of various matters of punctuation, format, and citation. If you don't have to follow a particular style guide, you can adopt one that you find compatible with your own preferences, or pick and choose specific style rules from several guides, or go it alone.

No one will be left clueless as to the meaning of vs in Rousey-vs-Holm if you decide to omit the period at the end of vs. I have seen numerous publications that drop the periods from lowercase abbreviations like vs on purely aesthetic grounds (like yours), though many more follow either AP or Chicago. My advice is, Dare to do it your way—but do it consistently.

I wouldn't use a period after vs even when it appears in a non-hyphenated format. The trend is to get rid of various unnecessary and extraneous periods, as in the change from Mr. to Mr as the abbreviation of mister. (And Dr for doctor.)

Who's playing the game?

It's St Louis vs Boston.

Note also no period after st, although this may be too avant-garde for some.