Mysql – Export massive MySQL database, from a shared level hosting account

My guess is that it's a script time-out issue with phpMyAdmin, or perhaps a file-size limitation of the underlying webserver.

Update
On a script-by-script bases, you can add set_time_limit(0); at he top and it will override the PHP settings for script timeout to infinite. Any value other than 0 will set it for that many seconds.

If you wanted to keep that "slave" SQL Server, I would suggest forgetting about replication just loading a daily backup onto the server, or fiddling with restore transaction log backups. The DBAs should have those handy, you would just have to convince them to help you get an automated restore of the production data onto the "slave" box. BUT, it sounds to me that you would rather that the slave box goes away.

I would look in detail at the "Change Tracking" and "Change Data Capture" features. From what I have read, "Change Tracking" identifies rows that have changed in monitored tables by providing a list of primary key values. "Change Data Capture" provides more information, including capturing actual before and after values, for full auditing. The hand-wavy blurb is that you would read from SQL Server tables that describe what's changed and then update your mysql database. No fancy replication, or agents, or anything. Here is MS's landing page for those features.

It sounds to me that you could just use Change Tracking and overwrite whatever is in mysql with the currently-good value from the SQL Server. I would expect that Change Data Capture would put more load onto the box than Change Tracking.

As far as the daily 3GB extract goes: I have seen "over-provisioned" servers with poorly performing SAN storage. Having lots of cores doesn't fix every problem. Overnight periods often have other stuff going on, like backups, defrags, other extracts, large report runs, etc. To me, "internet retailer" implies that you could have traffic at any time. The DBAs might simply be wary of adding any load that the don't have to.