Considering the root logic of the system, reaching back into RM2, where spells always go first, the answer would be no, the spell always goes off first, with order of initiative only modifying order of melee, but casting phase always preceding missile phase. That was the way the spell worked when originally written. . .this isn't the RM2 board though.
Using an activity/time method, likely the caster would be able to see the archer in process and get an instant off before the archer could complete the shot.
More appropriately on topic in this sub board. . .Stepping forward into RMFRP. . .If the archer wins initiative and declares a snap action, they should shoot before the caster can cast. . .if the archer chooses to fire on normal or deliberate, and the caster snap casts, they can get it off first, despite losing the initiative. The round is more complicated than just initiative. . . .snap/normal/deliberate declaration takes precedence, then initiative order.
You could rule as GM back to the days of RM2, and the spirit in which the spell was originally written, and let it go off first.
OTOH they did modify the spell in it's conversion over to RMFRP, and the complaints about the fact that in RM2 a caster with a -20 quickness bonus who rolled as badly as possible on init could cast a deflection every round and get it off before an archer with a +30 Qu bonus who rolled as well as possible on initiative could shoot them. . .and repeat it every round while devoting their remaining activity to fleeing to cover on movement phase was part of the reason why the ordering of the round was changed in RMFRP to make it potentially possible to shoot or stab a caster before they could teleport away or blast the attacker regardless of initiative and relative speed.
Keep in mind, that by extension, if an instant bladeturn or deflection goes first regardless of initiative, it then makes sense that any instant spell does the same. . .which covers not only many defensive spells, but a horde of offensive spells as well, enough that pretty much any pure and most semis will have a spell attack that they can use to front load every combat and win initiative regardless of the roll. You're much more wary going to take a shot at a priest (unless you shoot them from ambush) if you can expect them to pop you with something like Blinding or Absolution before you could possibly shoot them.
I played enough RM2 where casters always went first, and had a great time. . .so I won't say it'll ruin your game, if you like it go with it . .I'm not telling you not to do it, just warning you that from my experience, it will shift power toward casters and make melee / missile characters less effective . . .be prepared for any non caster PCs to get frustrated about it. (And for people to start just killing casters from surprise ambush rather than risk an open confrontation where the caster has such a dramatic speed advantage.)