We consider three-dimensional (3D) lattice Abelian Higgs models, with compact U(1) gauge variables coupled to a doubly-charged $N$-component complex scalar field (CLAH). We focus on their phase transitions between the disordered-confined (DC) and ordered-deconfined (OD) phases. When they are continuous they belong to the 3D Abelian Higgs (AH) universality class associated with the stable charged fixed point (CFP) of the renormalization-group flow of the 3D AH field theory, or scalar electrodynamics, describing $N$-component complex scalar fields minimally coupled to a U(1) gauge field. This CFP exists only for a sufficiently large number of components, i.e., $N \ge N_d^*$, where the integer $N_d^*$ depends on the spatial dimension $d$ (for example $N_4^*=183$). To estimate $N_3^*$, we look for the minimum number $N_{\rm cL}$ of scalar components of 3D doubly-charged CLAH models developing continuous transitions along their DC-OD transition line. For this purpose, we present finite-size scaling analyses of Monte Carlo simulations for $N\in[4,10]$, up to lattice sizes $L\approx 100$. The results provide evidence of continuous DC-OD transitions for $N=10$, and weak first-order transitions for $N\le 7$. They are not conclusive for $N=8,\,9$. Therefore, we estimate $N_{\rm cL}=9(1)$.