Welcome Jason!

>   I've been looking for a mesh network for my home to ease the trouble with
> flakey and various APs from different vendors.  One issue I continually run
> into when shopping for a mesh setup is that many of the commercial ones
> want to replace your router, take over DHCP, etc.
>

A LibreMesh deployment will give you a fully functional mesh network.
LibreMesh nodes are not only APs, but routers too. So, in principle, you
don't need anything else than LibreMesh nodes.

> With LibreMesh is it possible to continue using my own router?  I have a
> Cisco ASA that I am somewhat partial to and I don't want to stop using it.
> I couldn't find anything (obvious) in the docs on how if this scenario was
> possible.
>

As far as I know the Cisco ASA cannot participate in the network as a
mesh node. Maybe you could use it as a perimeter firewall or as an edge
router?

Maybe it is possible to let the Cisco ASA be the DHCP router of the
LibreMesh network, probably someone else here in the list knows better
than me.

But anyway, have you considered the invaluable benefits of getting rid
of evil technologies and jumping into the Free Software world?