OP, some of these answers are flat wrong. These mods are not high weight. You are more likely to hit +skills than you are either of these mods.
https://poe2db.tw/us/Staves#ModifiersCalc
T1 Spell damage = 50 weighting / 41400 total pool = 0.121%
T1 Phys Dmg = 50 / 41400 = 0.121%
+7 skills = 100 weighting / 73000 = 0.137%
If it were me, I'd cycle abyss essence suffixes until +7 phys. Would also use perf exalts with omen of dextral crystallization to get free annuls on the suffix and hope that a perfect exalt hits +7 phys.
If at any point you hit your 3rd perfect affix, you can desecrate (without revealing) to fracture and protect a mod.
If you hit the +7 with the perf exalt rotation, your path forward is really easy. Either desecrate the prefix for your gain as fire you said you preferred, but you'll have a harder time whittling suffixes for t1 cast speed and t1 crit.
Or, if you are willing to take any gain as %, you can slam/whittle the prefix for t1 gain, then whittle for either t1 crit or t1 cast speed.
Iirc whittling is a little cheaper than chasing t1 crit with the desecrate but can't remember exactly. If you do hit T1 crit with the whittle before T1 cast speed, desecrate cycle for T1 cast speed. Don't try to whittle for it.
Lastly, and probably most importantly -- if you don't have a lot of currency, or any of this sounds overwhelming, you are probably better to just sell the base as is and just buy a +7 phys staff which should be relatively cheap.