I want to move my current family to a bigger house, but one sim is about half way through the Perfect Garden LTW right now. Is there a way to bring the plants with them when they move?

If there is a way to do that I'd like to know too, but I don't think there is, unfortunately. However, if you harvest right before moving and replant after, your sim can pretty much continue where they left off anyway.

Or simply go back to the old lot with your sim every day and treat it like it's still their garden. Works as well, unless all the plants are indoors.

I also don't think there a way to move garden plants. Sorry. But I do what WoolySmurf does. I just generally have all the stuff in my fridge. I keep 5 of the best of everything in there so when I move I can just plant one of each of them. Or the same when the plant dies. It's the best option since fridge contents often get packed. Although I often use Debug to bring some of my plants back to life, ie Fruit trees!....Because it's stupid to me that fruit trees die and go barren every year.

FYI: I'm not sure about which mod actually allows this, so forgive me, but I think it's the 'icarusallsorts.ServeMeals.' that allows you to stock the fridge all at once. Then you have options to pick best quality of all ingredients either 5 or 10 of each item, depending on how much you have in there. I use it often and then sell what is left over.
BTW: If someone else knows I'm wrong about the mod, please feel free to correct me. I'm not really sure since none of the mods I have say anything about the fridge specifically.

I use buzzler's "Box of Seeds":
https://modthesims.info/d/415450/the-box-of-seeds.html

That way, my perfect life fruit don't accidentally end up in some random breakfast pancakes. Of course, the game loves to delete consumables when you move.
If anything goes wrong, I would probably just put buydebug plants on the lot and use Debug Enabler to bring the plants back to the quality in which they were before. The LTW just counts what is on the current lot. Doesn't matter who actually does the gardening.

If you're moving within the same neighborhood, I'd do as WoolySmurf suggested and buy an additional house in the neighborhood.
That way you can tend the old garden and it will still count towards your perfect plant count.

Once your sim has completed the lifetime wish and the plants on the original lot are barren, you could sell off that lot.

I always put 5-10 perfect produce in a chest to be on the safe side, so you always have perfect quality to replant if needed.

If you click on the dirt mound of the plant, you can use DebugEnabler to "move to family inventory". You can then place them at your new home (may possibly need to have Move Objects On to place them... don't recall off the top of my head).