It's not clear to me that you have a cycled tank to get seeded media from. I'll assume not.
You have the right idea: house your fish in the larger tank and do water changes enough to keep ammonia well below 0.25ppm while you cycle the filter in a spare tank or even a bucket. Read about "bucket cycling" here:
This is easier on your fish, but it takes about the same amount of time if you don't have filter foam or gravel with nitrifying bacteria in it.
If you have seeded media, put it in the new filter or in the flow around the new filter and use your fish as an ammonia source. Monitor carefully and change water to keep ammonia <0.25ppm.