Ping only tells if the machines can talk to each other - which at least confirms that the network connection is fine. However, to check if the client and server applications are properly exchanging packets you need to run a packet sniffer (I used Wireshark) to monitor the UDP packets that are sent and received by the client machine, while you are running the FRI application. These packets are also timestamped, so you can tell if the client is taking longer than the set sampling period (10ms in your case) to respond and hence confirm the cause of poor connection quality.
I was able to go up to 250Hz (4ms sampling period) with plain out-of-box Ubuntu running on a slightly old 4-core intel processor, beyond which I would start getting connection quality related errors. So, I believe a 10ms sampling period should work just fine with whatever flavor of linux you go with.
I would say try and make sure FRI works with regular linux before you start going down the rabbit hole that is real-time linux :). I use a real-time linux because I need to run FRI at 1000Hz, which I could not get with regular linux. Also, no guarantee of hard-real time performance with regular linux if you care about such things.