Consequences of turning $\oplus \text{SAT}$ into few satisfying assignments
The polynomial hierarchy would collapse to the fourth level (indeed, the third, see below).
Proof. First, we get that $\oplus P\subseteq \Sigma_2^P$. This is because a $\Sigma_2^P$ machine can use its $\exists$ quantifier to guess a polynomial number of assignments and then, using its $\forall$ quantifier, verify that this list contains all of the formula's satisfying assignments. By Valiant-Vazirani we have $PH\subseteq BPP^{\oplus P}$. We have $BPP\subseteq \Sigma_2^P$, so we get $BPP^{\oplus P}\subseteq {\Sigma_2^P}^{\oplus P}\subseteq {\Sigma_2^P}^{\Sigma_2^P}=\Sigma_4^P=PH$. $\square$
@Emil Jerábek gives a stronger argument; the polynomial hierarchy collapses to the third level. This is because counting assignments can be done in $P^{NP}$, which is better than the $\Sigma_2^P$ method described above.
Proof. We get $\oplus P\subseteq P^{NP}$, because a $P^{NP}$ algorithm can count a formula's satisfying assignments, if it is polynomially-bounded. (algorithm below). Consequently, we get $PH\subseteq BPP^{\oplus P}\subseteq BPP^{P^{NP}}=BPP^{NP}\subseteq \left(\Sigma_2^P\right)^{NP}=\Sigma_3^P=PH$.
Algorithm [For counting satisfying assignments in $P^{NP}$].$\quad$ Given a Boolean formula $\psi$, query the oracle. If the formula is not satisfiable, return. Otherwise, find a satisfying assignment $x$ using $n$ more queries to the oracle. Remove the assignment, i.e., construct a new formula $\psi^\prime:=\psi\wedge \neg x$. Repeat this procedure until the oracle returns ``not satisfiable''. This happens exactly after all the satisfying assignments have been &quot;removed&quot;. This procedure therefore runs in time polynomial in the number of satisfying assignments, which, by assumption, is polynomial.
The inclusion $\oplus P\subseteq P^{NP}$ is obtained by simply returning ``yes'' at the end of the computation if and only if the number of satisfying assignments is discovered to be odd. $\square$
Perhaps there are other consequences; I hope others can give some.