Well,
First I am not sure include() handles URLs, but maybe it does...
... Still if *.php files are processed by the preprocessor, you don't get the file included in your script, but the result of the script. Which if this file is only an included file (with class, function definition), it returns nothing!
So you don't get the definition of the function included...
Solution (but that is messy!):
rename the file to be included to some extension that isn't interpreted by the hosting server first (like .inc, as I do for my include files...)
But again, I place my includes in some directory that is not accessible from the outside (like somesite.com/includes for includes and somesite.com/htdocs for public). And if you do so too, you won't get access to your file from another server (client in this case)...
Be aware that if there is a passwd or something (like to log on the DB backend) it would be visible by anyone... So you should only allow your other server's IP to be able to access the (not being processed) php include file directory.