Rather than SMTP, I believe Exchange/Outlook prefers to use it's own protocol (EAS or MAPI or something like that).
Sorry my memory is hazy as it's been a long time since I did this stuff but I'm thinking along these lines...…
Take in a managed switch that can do port mirroring and your own pc/laptop with wireshark setup.
Now lets say you have an 8 port switch. Stick cable from office lan into any port, and for example, put one of the VOIP phones on port 1, and setup switch to mirror port 1 to port 2.
Now with your laptop you should be able to observe traffic to/from the phone. Make a couple of test calls and you should be able to figure out the DSCP (QoS) settings in use for VOIP.
Now unplug the phone from port 1 and plug your client's machine for testing into port 1.
Run wireshark and observe what happens when you try to send the large email (probably heaps of TCP retries).
Now manually set QoS on the test PC NIC (using same settings as the phone) and redo the test. If it all behaves nicely now you know QoS is throttling your email traffic.
This all assumes the phones and desktops are all on the same lan/vlan, and QoS is using Diffserv.
Or another method, we are both assuming QoS has been setup to prioritise VOIP and this is playing havoc with emails. If you can have sole use of the network again, try other protocols. Any way you can test bandwidth to prove bandwidth the office is receiving is nowhere near what has been sold to your client (and the key point - that a significant amount of your client's traffic is being rejected by the building manager's equipment). Once you have good proof of this, make a nice bill, and get your client to forward this to their building manager with a notice that internet must be fixed at their expense, the quicker this is resolved, the lower the claim will be for breach of contract (unless contract specifies it is a contended/restricted service in which case they are shafted and might as well look at getting their own internet).
Or you could just play naughty and try to make all your client's traffic get prioritised as if it was VOIP. If it's Diffserv that should be a piece of cake. I've seen some implementations of VOIP that prioritise certain ports, just bodge the settings at the client end, then at your end setup a mail proxy and setup your router to port forward the traffic to your proxy on the correct port.