Remote Logging vs Local Logging – Does Remote Logging Stop Local Logging?

No. For one, the handoff is asynchronous within the local operating system. syslog libraries and the local syslog daemon will either accept the message and fail to deliver it or fail-fast, but either way your app won't hang. Secondly the network protocol is (by default) udp so even if your application blocked until the packet got sent out, it would go out instantly and return control to your app regardless of if it actually makes it to the collecting host.

When people think of remote logging hanging things in *nix land its usually because they were logging to an nfs mount, which most certainly can cause hangs. Syslog, you're good.

Put a packet sniffer on the Splunk host, set up a capture filter to capture only packets that originate from the ASA, start a capture, when splunk stops "seeing" data from the ASA look at the capture and see if traffic is still coming in from the ASA, if it is then the problem is with Splunk, if it's not then the problem is with the ASA or the network between the two.