Spotty connection to my web host, but they say it's not them. Help?

I'm having the weirdest issue with my web host and I was wondering if anyone knows of a way I can test the connection, to figure out the problem, because their Level 2 techs are unable.

So at work, we have a website that is hosted on a web host. The ISP I'm using at work, is the same one at another location. At that location, it works flawlessly... never had an issue ever. At work for the last three months I would lose connection, and then it would magically start working again 5 minutes later. This all stopped two days ago, where now it only works for a few seconds and then stops. This does not happen at the second location (again, same ISP). We do have a firewall, but it's not blocking anything at all. If I switch our public IP address, the website starts working again magically for a few minutes... and then it stops again - sometimes for hours at a time. Switch IP? Works again for a few minutes. We have a FortiGate firewall.

My web host claims the problem is NOT on their end, and that they've whitelisted my IP. They don't know about my secondary IP address which I test with, but it has the same exact result.

My question is, how in the world can I figure out what's going on here? I've even asked them to migrate my site to another server which solved the problem for about 90 minutes, then it went back to its old self again. Are there tools that I can use to connect to a webserver and test connectivity? They have cPanel - is there something from within there that I can use to test? I want to make sure it is on their end, which I suspect it is.... but I don't know anymore.

To clarify, everything except for ICMP fails when I'm "blocked". FTP, Web, Email, you name it... I can't access it... but pinging works just fine.

Thanks!