Yes, you can run the bigger filter in your tank instead of the small one. Just so we don't get confused, I will just talk filter pads ... "old" means the ones that is currently in your tank, "new" means any unused filter pad.
OK, you replace your current filter with the big one, put both old & new filter pads in so both get water flow. <Make sure nothing dries out in the change process!> Run this filter for 4-6 weeks. For the first week or so, check your tank's levels to make sure you have not disturbed anything important causing a mini cycle.
After the new pads are seeded, you have 2 choices:
1. Immediately add fish. Because the new pads has low bacteria load, it will not be able to support many fish. <This is especially true if you have a small tank, & you want to start a bigger tank with it.> You can at most add 1/2 the bioload (of the old tank) to the new tank at the start. It will be much safer to start much lower and add fish very slowly (one or 2 every 2-3 weeks), meanwhile monitoring the water parameters daily & doing pwc's for the inevitable mini-cycle. I have done this before & mini-cycle generally lasts a week or 2, but normally don't give too high a peak & fairly easily managed with some pwc.
2. Build up the bacteria with fishless cycling. This is the safer method. You take the seeded pads & run it in the new tank. Then you add ammonia to feed the bacteria to grow it out so it will support a heavy bioload. The process takes maybe 1-2 weeks. You know you are done when you add 4 ppm of ammonia to the tank, and all of it is gone to nitrates the next day. At this point, you can add all the fish you want at once. Keep an eye on the levels for a few days afterward, but if your tank can handle 4 ppm of ammonia, it should not show any spikes at all.