I can't say whether it fits your particular threshold for "wordness", but psst has itself seen use as a verb:
I was just rounding the fourth-floor landing when Claudia pssted
at me from my mother's door (Dale Peck The Garden of Lost and
Found, 2015)
“Oh my God.” Someone pssted next to Bree, shoving their phone into
their neighbour's lap. “Have you SEEN this?" (Holly Bourne The
Manifesto on How to be Interesting, 2014)
This happened to me on more than one occasion, being “pssted” at, and
all I could do was smile, nod my head, and continue with my novice
picking. (Russell Zanca Life in a Muslim Uzbek Village: Cotton
Farming After Communism CSCA, 2010)
It was not just men psst, psst-ing at women. Men psst, psst-ed at men.
Women psst, psst-ed at men. One day an older woman psst, psst-ed to
ask if I would hold her arm and help her cross the street. (Lea
Aschkenas Es Cuba: Life and Love on an Illegal Island, 2006)
These examples are fairly representative, but not exhaustive. There are also examples from the greater web, including a pssting hashtag.
This does seem to be a relatively recent usage; I didn't find any reliable hits in Google Books for pssted or pssting before the 21st century. There also doesn't seem to be any standardization of exactly how to write these—in "dubious quotes", hyphenated, or just plain verbed.
With all of these caveats in mind, I think the meaning is quite clear, especially in print, and the usage does seem to have at least a toehold, so if it works in your particular context I think it would be fine.