Default route (internet bound) over EIGRP stub overlayed with 3rd party managed MPLS

I feel silly for even having to ask this, but it's been a little while since I played around with routing in this way...

Long story short - we are getting rid of our Australian data centre, which has direct BGP interface with our MPLS provider (Centurylink). Instead we are planning to route the one site without a dedicated internet circuit ("Moo" in my diagram) via a site in Sydney for their internet connectivity. We also will need to configure a failover for all the other sites we have in Australia to route interent via MPLS toward Sydney.

I've mocked up a like-for-like environment in GNS3 (minus whatever the hell Centurylink do in their MPLS cores, supplemeted with some basic BGP, no VRFs).

https://i.imgur.com/UWzrcpC.png

Shoving an overlay route on it doesn't work, i.e.:

ip route 1.1.1.1 255.255.255.255 <sydney core switch IP> 

I am not 100% sure if Centurylink have some safeguards to stop me advertising a default route directly from Sydney... Unfortunately we don't have time to mess around with asking them to make changes in their configuration.

Notes:

• I have read-access to the Centurylink CPE's but not PE's, so I can't make changes anywhere in their environment
• The new Sydney site where I am planning to break out internet only has an EIGRP peering (which is redistributed into Centurylink BGP on their CPE)

I am thinking this is a relatively easy one, and I am just missing the trees for the woods... any help is appreciated!