Why will 26 1.2W 220V LED lights not work with an 800W inverter?
This is almost certainly from using a "modified square wave" - MSW - inverter rather than a "true sine wave inverter" - the power conversion circuits in AC LED fixtures expect a sine wave input, and the way they behave when fed MSW input is upsetting your inverter.



One possible solution, short of "buy a new inverter" (a rather expensive proposition, especially for true sine) would be to feed the light circuits (only) though a small isolation transformer. The inverter power would feed into one side, and you'd feed the lights a much better approximation of a "true sine wave" from the other side. It should be a bit oversized for the load, since the MSW input will cause some heating of the transformer that a normal sine wave would not. However, the load is so small that almost any isolation transformer will be a bit oversized for it. Find the minimum size by looking at the VA (not W - power factor matters, a lot, here) rating of your lamps, and then run a fudge factor for MSW input of about twice that rating. 

Unless the VA is absurdly bad for a 1.2W device, something like this for roughly $50 ($40.76 plus shipping) will probably work. You'd also need to mount it in a safe enclosure, provide circuit breakers or fuses, etc...