Both of these are very expensive in low-volume.
(This may something you already had in mind, summarized as "high-end FPGAs;" I just wanted to add some interesting color some folks might not be aware of.)
http://blog.zorinaq.com/asic-development-costs-are-lower-tha...
At the very least SpaceX would test the devices to see what effects the higher radiation would have. However it's not unreasonable to just have the software detect and reset itself, if a high energy event is detected. There are going to be a lot of these satellites and a lot of redundancy in coverage, so if one is down for a few minutes it's not that big of a deal.
Speaking of which, there's a neat project that sneaks into a particular Intel chipset and extracts the CSI (Channel State Information) from the wifi hardware so you can actually do things like estimate which frequencies are being attenuated [1]. It's out of date, but still shows the kind of live, complex, signal analysis that is possible with commodity-priced hardware.
In short, I think it is certain that a base station can be built for <$1000, and likely that it can be built for <$150, although without knowing the precise specifications, it's hard to say. We did a case study on this in my antennas course in grad school 3-4 years ago, and our ending numbers on our prototype designs (not fully built and tested, of course, but simulated and spec'ed out with real-world pricing) were around $150. I am certain that our design as it was would not work, but something similar to it likely would. ;)
[0] https://en.wikipedia.org/wiki/Beamforming [1] https://dhalperi.github.io/linux-80211n-csitool/