Not quite, SSA form makes only the colouring pass polynomial time, optimal choice of spilling and coalescing is still NP-complete. cf. https://hal-lara.archives-ouvertes.fr/hal-02102286v1/documen...
I got the impression that expensive register allocation is not an option. A 1% improvement in register allocation that results in a 0.1% local performance improvement and a 0.002% program performance improvement may be considered not to justify 1000x local compile time, but for frequently run stable-ish software the compile time and compiler development/maintenence time might be justified.
(I am not a programmer, but I do like thinking and learning about such things.)