Move juvenile to brand new tank from older small one?

Just received a juvenile, almost 4in/10cm long, they're in a small transportation tank - only about a gallon or two, I think it'sjust one of those little goldfish tanks. The ammonia in there is obviously a bit high - it's at 2ppm, nitrites are 0-0.25. pH is 7... The newer tank is uncycled, but colder, 20gal, 68-70 deg F (currently setting up fan), and obviously has no ammonia, nitrites, or nitrates. I have a larger, long-established 50gal tank with a pleco, but also a small breeding divider to hang on top. Both have the same canister filter. I seeded the new tank with a few plastic plants.
I know I need to get the axie into the new tank as soon as possible, but the issue is the lack of cycling in the new tank. Could I take some filter media from the old tank to speed up the process? Would it technically be cycled then? The old tank is at 77-79 deg F, would the sudden temperature change kill the bacteria?
Would it be best to move the axie to the new tank now? I could check the ammonia levels everyday and do water changes, but would that be stressful?
Or should I put them into the breeding partition or leave them in the transportation tank until the new one is cycled?
Thanks

In my opinion, you should put them in the new tank now (carefully acclimating them, of course.:happy

Do not put them in the old tank - the temperature is too high.

Yes, you can seed the filter on the new tank with media from the old tank, and that will help. However, it will not provide an "instant cycle". As you suspect, the difference in temperature will favor different good bacteria. Plus, the bacteria will take some time to colonize the rest of the filter and other tank surfaces. So, keep a close eye on on the ammonia, nitrites, and nitrates.

Good luck, and enjoy your new friend. :happy:

Thanks! What do you recommend as the best acclimation process? Just placement in a bag of the old water in the tank to slowly match the temperature? Thank you for your help!

Equalize the temperature, but also, gradually add in the new water to equalize other the water parameters. Dump some of the water out of the bag occasionally to make room for more "new" water.

Start with, like, replacing 1/8 of the water in the bag with "new" water, wait for, say, 10 minutes, then replace, say, 1/4 of the water with new water. Keep doing this, making the amount larger and larger, until the temp is the same and the water parameters are also the same. Note: if the pH or temperature of the "old" water is significantly different from the "new" water, do this process more slowly.