I don't think your getting a 414 from the Google API I think your either getting it from apache or the web browser.
Under Apache, the limit is a configurable value, LimitRequestLine. Change this value to something larger than its default of 8190 if you want to support a longer request URI.
Google Analytics API limits the number of dimensions and metrics you can send but there is no limit to how big your filter can be. (Assuming you are stringing them together using and
and or
a single filter value can be max 128 or you will get a 404 bad request error)
But assuming you keep tacking them on you will probably end up blowing out the limit set by apache. (Being that your not really sending the request though a web browser I wonder if you could set apache higher to handle the request. A web browser can max handle 2000 but this might not be an issue in this case because your not going though a web browser. But that will depend on if Google will even except a request that long.) I will have to test this.
Edit Post: No The request to get data back form the API is a Get not a Post. Post would be sending information to the server not Requesting the server to give you information back.