Any compatible trigger shifters for el-cheapo TD ONE derailleur?
You need a 6 speed shifter.  These are likely to be rare, so a 7 or 8 speed shifter should work, but will have 1 or 2 ghost positions which may be confusing.

Another possibility is to use a friction shifter, where the rider has analogue input and can put the chain where it needs to go.   

But from another point of view, you're throwing good money after bad.  A cheap bike is okay while they're still growing, but at some point you're better off saving for a good bike.

If there's a bike cooperative in your area then they may be able to assist with take-off parts or similar.