Sampling satisfiable 3-SAT formulas
There is a simple algorithm for V3.
I'll use the convention that there are $(2n)^3$ possible clauses,
so $2^{8n^3}$ formulas.
(This is just for simplicity - if you don't want all $8n^3$ clauses to be considered valid, it wouldn't affect the following argument.)

Pick a random assignment from $\{0,1\}^n$.
For each of the $7n^3$ possible clauses that are true on this assignment,
include the clause with probability $1/2$.
Each formula $\phi$ will appear with probability proportional to
its number of satisfying assignments.
The $m$-clause variant is similar: pick a set of size $m$
out of the $7n^3$ clauses.