Are there any words I can use to disambiguate "biweekly"?

Avoid biweekly altogether. Use fortnightly for "once every two weeks", and twice a week for, well, "twice a week".

Not everything has to be a single word, so don't be afraid to use more than one word when you want to use clear, understandable, unambiguous language.

If they have problems with fortnightly, use every other week, or let them into a little secret: fortnight comes from "fourteen nights", or two weeks.

Edit: From the comments it's clear many Americans won't understand or are uncomfortable with fortnight so, to be safe, use the aforementioned every other week.

Technically, semiweekly is the term that you're looking for. But if you're trying to avoid ambiguity, then go with something like "twice a week" like Hugo suggested. Too many people get biweekly and semiweekly confused.

I agree with those who suggested "fortnightly." If someone feels that it's antiquated or "odd," that is their problem! It's a great word. Besides the ambiguity of the words "bi-weekly" or "bi-monthly," I think that they are esthetically ugly and artificial words that detract from the English language. I like to keep my Latin and Germanic mixing to a minimum. Try "twice weekly," if "fortnightly" doesn't do it for you.

Strangely, although bicentennial, bilingual, and bipedal (among many other actual and imagined bi-prefixed words) would never be understood as referring to half- century, language, foot, etc. phenomena, biannual (or biennial) or bimonthly or biweekly (and probably bi-daily, if anyone ever tried it out on people) do elicit that interpretation (perhaps largely among the semi-, but certainly not the bi-, literate). Dictionary.com offers the following highly laudable advice:

"Since bi- can be taken to mean either “twice each” or “every two,” a word like biweekly can be understood as “twice each week” or “every two weeks.” To avoid confusion, it is better to use the prefix semi- to mean “twice each” (semiannual; semimonthly; semiweekly) or the phrase twice a or twice each (twice a month; twice a week; twice each year), and for the other sense to use the phrase every two (every two months; every two weeks; every two years)."

As for the claim that "every two" (or "every other") years/months/weeks/days beats biennial/bimonthly/biweekly/bidaily, it fails to meet the immediately-intelligible test that ought to govern all linguistic prescriptions.