"provided" used as a conjunction with the meaning "on the condition" is correctly used in such a sentence:
"we can cut the budget provided keeping the same low level of production"
more sense to me makes this phrasing:
"we can cut the budget provided that we keep the budget at the same low level of production"
thx
(1) We can cut the budget provided (that) we keep it at the same low level of production.
(2) We can cut the budget if we keep it at the same low level of production.
Yes, it's better, but note that the subordinator "that" is optional; also I'd replace the second "the budget" with the pro-form "it". Additionally, as in (2), you could use "if" instead of "provided" (though dropping the "that" would be obligatory).
In both (1) and (2), the underlined sequence is a conditional adjunct. We understand that if the condition is fulfilled, then the budget can be cut.
Note that "provided" is best analysed as a preposition, not a conjunction. Your example demonstrates that, because if "provided" really is a conjunction, then it would be possible for the conditional clause to be introduced by two successive conjunctions, "provided" "that", which clearly makes no sense.