So, I have this:
Now, what I want is a repeater that would work while keeping the unlit lamp on.
Yeah, that one right there:
I have this already:
...but the piston turns on and off repeatedly until the input is off.
It looks like you're running the power along a row of raised blocks behind the lamps. This normally works because redstone on top of a block will power the block below it, and that block will power blocks adjacent to it. This is why the repeater is causing a "dead" spot: the repeater isn't powering the block below it, so nothing is powering that one lamp.
Instead of powering a row of blocks beside the lamps, power every second lamp directly. Directly powering a lamp will light both the directly-powered lamp and the two beside it, as the middle lamp powers its neighbours:
Because redstone in a line won't directly power an adjacent block, to do this in a line you have to have the redstone "bus" one block distant, with singleton lines going into every second block. This conveniently gives you a place to put your repeaters:
Of course you can also power every third block instead of every second and it will work the same. If redstone is no object I prefer the redundancy and simplicity of two-block spacing, but three-block spacing works better when redstone dust is a scarce commodity.