Unrouteable management vlan means NTP doesn't work

We've added a couple of 6248s to our network.  For our needs, I've configured a few vlans, assigned subnets to them and enabled routing.  Since the management VLAN is unrouteable, I'm ignoring it (and the only ports on it are the ones connecting the two switches).  To do this, I had to configure a subnet and IP address Switching/IP-Addressing which doesn't conflict with anything we actually use.  Since we can access the switches via their IP addresses on the subnets we do use, that works.

My problem is that the 6248s cannot now synchronise time via NTP.  It seems that they want to connect from their management ip address, which will not work for obvious reasons.  I'd imagine that DNS is also not working for the switches and that Radius or TACACS authentication would fail for the same reason.  This isn't good.

If I use the CLI to remove the default gateway from the management vlan and add a default gateway to the routing table, will this fix the issue?  I'm a little sceptical, since the DNS and NTP servers are on one of the subnets assigned to a vlan.  If that won't work, what will?