Solve the unconstrained least squares problem in "one-shot", for example by QR or SVD (if not too big), if you consider that to be "one-shot". Then if the optimal $x$ to the unconstrained least squares problem satisfies $A x \le b$, it is optimal for the original constrained problem; if not, you have not solved the original problem in one-shot.
Of course it is possible to solve the original constrained problem in one-shot if the one-shot consists of querying an oracle which can solve such problems. The only difficulty from a practical engineering perspective is that you may not have such an oracle available - I don't happen to.
The question title mentions convex/non-convex. But the only the problem you displayed is convex, and ia a convex Quadratic Programming problem. Or it can be transformed into an equivalent Second Order Cone Problem (SOCP) via epigraph reformulation, which may be numerically advantageous from a robustness standpoint.
$min_{t.x} t$
s.t. $\|y-x\|_2 \le t, A x \le b$
Such QPs and SOCPs are not solved in "one-shot", as you seem to define shot.
If $y$ were a nonlinear function of $x$, then you would have a linearly constrained nonlinear least squares problem which might be non-convex.
There are specialized, efficient, and robust numerical optimization solvers available both for linearly constrained linear least squares ans for SOCPs.