You can technically have the tool-using ones keep their inventory with a log in it.
Fill their inventory with Stone except for the last slot, in which you put a Log, and their hand, in which you put the tool.
With this, if they burn their Log, they will stow their tool, grab a log, and then cycle back to their tool, leaving them back with a full inventory and a tool in hand. Then they'll do their task up until their hands are empty (and their tool broke), after which they'll go back to the tool storage to grab a fresh tool.
You can add an empty "Until (X) storage is not full" just after the "until Inventory Full" block if you want them to not bother doing their task if your storage is full for that item anyway. For example, I could say "Repeat Until Clay Storage is not full : (Nothing)" and the bot will just sit there waiting for there to be room in Clay Storage before they start digging again. Since waiting around like this doesn't use any fuel, it's okay that the program will wait there until there's room in storage.
When the bug is fixed that is making the bots consume their entire inventory, you don't have to prime them with stone anymore. They'll still have a full inventory of logs they won't be using, of course, but the time saved not having to hand them a bunch of stones first makes up for it.
Edit: Actually, there's some problems with this, because for some reason, burning a log that's at the front of your inventory won't actually shuffle the rest of the inventory down. Instead, it leaves the gap at the front of the inventory, meaning it messes up the cycling since when you store the item, it'll be at the beginning instead of at the end. So you have to use a full inventory of logs, and use Cycle item Up instead of down.