iPhone 5Ss
start from under $200, so it is a fair saving compared to the SE.
With regards to how it runs iOS 10/11, the processor benchmarks fairly similarly to the iPhone 6. As such, it's not likely to struggle for quite some time. Plus as the 5C/5 are still being supported with their 32-bit CPUs, the 5S is far from the cutting line.
OP, I'd definitely recommend the 5S.
That link showed a 5s for $240. Still a good savings over the SE, but not quite under $200.
The 5/5c are now the bottom of the heap, which means the 5s will likely be the lowest model supported on iOS 11 and is likely to be dropped in iOS 12, in fall 2018. And most models don't run great when they are the lowest specced model still supported (e.g. A5-based devices on iOS 9 are no treat).
All of which means I would expect iOS 10 to be the last version the 5s runs well, and iOS 11 to be the last version it runs at all.
Of course sometimes the pattern varies - the A5 devices got more updates than the A4 devices because that was single vs dual core, so a big jump. the A7-based 5s was also a big jump, so maybe it gets an extra year too, who knows.
Regardless, the SE is going to have support far longer, and be a better experience in the short term. But only the OP knows his own financial situation and how much of a stretch the SE would or wouldn't be.