Let me preface this by saying that I'm by no means an expert or even well versed with cabinet wiring and schematics.

I'm currently running the following set-up

SX Jazz Bass copy
Gallien Krueger 1001-rb II head (700 watts)
Genz-Benz XB2 4x10" cab (700 watts)
Pedals: Boss TU-2 -> Boss GE-7B -> MXR Phase 90 -> Boss ODB-3 (soon to be replaced by a Russian Big Muff)

One of the main problems I'm encountering is that either the EQ unit and/or the OD unit are putting a little too much strain on my cab. Since the head and cab are matched up perfectly, I feel like these pedals are pushing it over. I'm thinking that upgrading the power handling of the cab, effectively giving it a little more headroom, will solve it.

Here's where my main question comes into play. I know that replacing the speakers with higher rated ones will increase the power handling. Would I be able to possibly only replace one or two speakers? I really only need another 50-100W of power handling. I'm just not sure if having speakers of different ratings within the cab will be a problem.

Thanks in advance to anyone who can help me out with this.
It depends on the ohm ratings of the current speakers. Pull them out and have a look at the back of the speaker. You should try to get the same ohm rating as the original that way you don't have to mess with wiring except swapping the speaker. I'm not sure whether it's gonna help at all tone wise.