Exactly what are you doing? Modifying an existing speaker? Building a speaker? What's your question?
I'd agree that "bigger is better"
(usually) and I don't really consider anything less than 8-inches to be a "woofer", and certainly not a subwoofer. (I'm an old-school big-speaker snob!)
What? Do you have a crossover now? Or at-least a capacitor to block the bass from the tweeter?
Lots of considerations...
When you "randomly" change a driver it usually has more or less output and that messes-up the "balance" between the woofer & tweeter.
Typically, a bigger woofer doesn't go as high in frequency so there may be a gap between the woofer & tweeter. But they vary all over the place so that's not always the case and it depends on the crossover frequency.
They make "full range" drivers and that may work better as a woofer in a 2-way system with a 10-inch woofer.
Usually when woofers get bigger than 8-inches, you'll see a 3-way design.
If you lower the crossover frequency, of course the tweeter has to cover a lower-frequency range (which it may, or may not, be capable of. And when you lower the crossover point, the tweeter has to handle more power.
In a properly designed speaker, the box is "tuned" to match the woofer. Any time you change a woofer it's going to change how it interacts with the box. (Ported speakers are more critical than sealed boxes.) A larger driver usually needs a larger box or else-the tuning/resonance goes-up and the bass rolls-off more (or more steeply).