I don't tt, but I rather wait instead of doing all at once, it just takes away the fun. But if I have a second town, I'll set it to southern and catch new bugs and fish there.
I catch bugs and fish as I see them through the month. As for fossils I only need one and even after TTing back and forth, back and forth I cannot get that last fossil. I actually bought one I needed and another I got for free since they weren't spawning either. The emptiness of the art/statues was bothering me so much I'm currently rerolling Redd these past 4 days (found out it will take around 2 years to complete? yikes), I'm so close to finishing but it's been a pain and I feel like giving up every time I get something I already have but at least I've gotten a good amount of nmt by selling them on nookazon for super cheap unlike ridiculous prices I've seen there. I did purchase a valiant and gallant statue though like i don't know why they haven't spawn either
There’s no wrong way to to it. I personally don’t find it fulfiling to trade for museum items though. I like taking my time and finding it all on my own and on my island, not a friend’s. Maybe I’m weird ha.
I usually wait for my island to spawn the seasonal bugs and fishes. In NH, the museum grows along with the amount of donation, so I really enjoy seeing how the museum develops over the time.
I time travelled mainly to get DIYs and experience the various seasons. Catching bugs and fish is secondary, as I can always travel to the month I need to get them.
There was one point where I made a Beetle Farming Island (similar to the Tarantula/Scorpion method) to catch all the beetles just because I thought it was a fun challenge and I like beetles.
My museum collection shows this as I have all but one bug, missing about 15 fish ( I don't like fishing), have all fossils, about half the art.
I've never finished my museums in past games. I don't think I'll finish it this time around either but I finished my fossils yesterday! I was cycling out villagers so that helped a lot with finding fossils. Also, mystery island hopping has helped too.