Щ is indeed read as "sht" in Bulgarian, but in Russian it would be read as a softer "sh" or mix between "sh" and "ch". If anything, maybe it can just be dropped in favour of plain "sh"?
From Ukrainian it gets turned into "shch". Which IMO sounds about right. Mainly because sh and ch are shorter sounds in English (ch can also be a little hard), but shch gets the point across.
This was also the historical Russian educated pronunciation; it only changed by the end of the 19th century.
Speaking as a Russian, I think that "shch" is close enough for an international conlang. It's not a sound that's particularly easy to learn or explain.
That's an option, but the big question is about "where it means sht".
Considering we're talking about constructed languages, whether "it's supposed to read `sh`" or "it's supposed to read `sht`" is defined by the author of the language.
On top of that, even coming to a conclusion on what it's supposed to read like means extrapolating it from present-day languages or from the now-dead languages. In a lot of cases, different languages will just have different readings. I.e. consider these words:
Plus there's the weird separate word [BG "sht" щателен - RU "tshch" тщательный].
Barring the loaned words from mostly Germanic origin, it looks like one language just sticks to "sht", while the other prefers "shch" and considers the "t" redundant. The interesting exception is the "що - что" pair, which is seems to preserve the T sound.
Ditto for я and ю.