There's a way to formulate an easy version of this question, which I could partially answer. (I'll make it CW, so people who know more about lattices could complete it.)

Question. Which Lie groups have a dense subgroup, such that this dense subgroup is a (sequential) filtered colimit of finitely generated lattices?

If Lie group is nilpotent, then there's such a sequence of subgroups if and only if its Lie algebra admits a basis with rational structural constants. This follows from the works of Malcev; having any lattice at all is equivalent to being Q-algebraic nilpotent group.

If Lie group is semisimple, then (as far as I remember) every lattice is contained in a maximal lattice, so there's no such 'approximation'.