I think you're going to have to speak to the sysadmin.
Obviously the sysadmins there control the network heavily, and they may not want you pushing the rules. If download is limited to 3Mbps, check the upload speeds; they may be much slower. Your transfer might take several days if you stick to network policies.
I assume that the file you want to transfer is a large video file or other data which can't be compressed without degrading quality, otherwise compression would be your best friend here.
Assuming you actually had 5Mbps upload ability (which is probably not the case), 44GB transfer at 5Mbps, divided by 8 bits, which is about 0.625 MByte per sec. Times 3600 seconds (one hour) that could transfer at most 2286MB. After 20 hours, 45720MBytes, or about 45Gbytes would have uploaded.
Assume some 10% speed loss, and that would be about 22 hours (if everything works smoothly and nonstop) and you monopolize the network for that much time, so you probably need to use a decent file transfer software.
You could split it to 10 DVDs, then ship by Fedex or UPS, but that might take about the same time.
Since the other side has 50Mbps download, they can download much faster, but that wouldn't matter much.
Another option might be Adrive.com will allow 50GB backups for free, and allow you to share it.
If network policy allowed access to www.megaupload.com (which I doubt) you could sign up and upload it all at once. Free account, but you will need to split the thing into some 50 small pieces, use the megaupload file manager, and upload them all there.