You fear the browser will time out? Don't use a browser for this. If you don't have access to a login shell on the host with the existing database, use a machine where you do have login shell access and see if running a mysql client on this other machine will let you connect to the database host -- a command line would be something like this to put the database into one file on the machine you're logged into:

mysqldump -h existing.db.hostname -p -u dbusername dbname > full_db.du
+mp
[download]

The "-p" tells mysqldump to prompt you for the db account password. If the existing db is in use while you do this (inserts and updates are being done), you'll want to check the manual for mysqldump about additional options about locking things while the dump is in progress, so you get a coherent set of data.