Some facts about the system would be useful.
In what country is the call center IVR? The agents?
From what countries do you take calls? Is there an access number in each, or do the callers make international calls? If the latter, to which country? What kind of access numbers (geographic, toll-free, mobile, national VoIP)?
Approximate size of operation (number of agents, calls per day, minutes per day)?
Why do the access numbers "call forward to another number that hosts ..."? It would usually be more reliable, less expensive and better quality to have the access numbers connect directly to the switch hosting your IVR by VoIP.
When callers have trouble, what goes wrong (error announcement, no answer, no audio, IVR answers but DTMF doesn't work, etc.)? Anything get logged for failing calls? How do the complaints come in? If by phone, on the same number?
Who are the callers (consumers, businesses, agents or employees of the company)?
Does the company have a physical presence in the countries from which they take calls? (You could put a GSM gateway there for test calls.)
1. Depending on the answers above, something like »
www.ebay.com/itm/GOIP-1- ··· f917fbd0 may be suitable. You'd need one in each source country. If you wanted to test from several operators, you would need either multiport gateways, or gateways that could use an external SIM bank.
2. I'm not aware of an external service, but you could write a script that runs on your PBX or other server that triggers test calls. They could dial a hidden IVR option that just hangs up, but you would process the call log to confirm that the option was properly reached.
3. Jitter would normally just cause poor voice quality. If you're stuck with inband DTMF, it might cause incorrect options to be selected.