The short answer: I don't know. I've never tried it, and it would be pretty hard to run a quick test as we are able to contact both.
Follow on question: is it the "ping" protocol/port/program that's disallowed, or any communication to a host using the fully qualified name?
A second follow on question: Are your machines set up to automatically add the remainder of the name (e.g. "DNS search domains") include mycorp.com, laptops.mycorp.com, etc.
A third follow-on question: Can you edit the hosts file on your machine(s) to create the mapping between IP address and FQDN? I'm guessing this is locked down since they sound like they have things locked down pretty tight.
A fourth follow-on question: Does your company provide a means of putting static DNS records in place and have exceptions to this policy? I can't imagine that you haven't already encountered other network applications that require a FQDN record in DNS to work correctly.
