Water pressure from a water butt can get through a DI but it is very very slow and it does need quite a bit of coaxing now and again. So it just is not worth the bother at all.
I have done it in emergencies, infact my system is set up so that I have a 300GPD RO then a small 250 ML DI sitting on the RO going into a water butt. I have a big DI for sorting it out if for some reason the water inside has gone above 001 TDS.
The 250 ML DI canisters are a pain as you have to keep filling them. However, I find it easy to flush my system at the same time as doing the refill. So it sort of solved two problems (stopping high TDS going through the DI at flush time and not having to wait an age for my DI water). I flush as soon as my water butt reaches 001.
Too be honest, go and buy a handheld TDS meter meter this week, then get some reading from the tap. As any advice is pretty much a waste without figures.