Wireless bridge interferes with host wifi

Solution 1:

Having the same subnet or essid is of lesser importance - no devices need to talk to one another

Having the same ESSID isn't about letting devices talk to one another – it's about letting them seamlessly roam from one AP to another as you move around (before the old signal drops to unusable levels).

but arp -a (which I believe should show which external MAC resolves to which IP)

No, it shows the opposite. ARP is used to resolve IP addresses to MAC addresses, not the other way around. (Also, there is no such thing as "external" MAC here.)

never changes for 192.168.0.1. Should not this change if I connected to the Asus?

No. Bridges are supposed to be fully transparent at MAC layer. If the entire network is bridged, then it is completely normal that 192.168.0.1 will resolve to the same MAC address everywhere.

Wi-Fi is one big exception to the above rule of bridges being transparent, but in your case it goes in the opposite direction – it's the IP addresses of "Net B" that will appear to have the Asus' MAC when resolved from Net A.

Your Asus, being an ordinary Wi-Fi client of the LTE modem, is not allowed to send packets "from" arbitrary MAC addresses – instead it has to rewrite all outbound packets so that they appear to be sent from the Asus' MAC instead, and it has to spoof ARP in the same way. So devices on "Net A" will always see the Asus Wi-Fi MAC for any IP address that's on "Net B" (due to Wi-Fi bridging limitations). In addition, this rewriting seems to completely break IPv6.

However, this limitation only applies to Wi-Fi clients, not access points, so any device on both nets should see the correct MAC for addresses that are on "Net A". Your LTE modem's MAC address should not appear to change.

If I power off the ASUS I can again connect to "Net A" without issues, meaning it seems to interfere somehow with the setup of the LTE modem

I do not have an answer for this, though I do have a guess. (Can't do much better if you have no means of testing anything else but Wi-Fi...)

As I mentioned above, Wi-Fi clients are an exception to bridging... but there's an exception to that, too – the bridge client and the AP can agree to use the so-called "WDS bridge" or "4addr" mode, which allows for 100% transparent bridging, preserving MAC addresses in both directions.

So my wild guess is that the ASUS tries connecting in the "WDS bridge" mode and this makes the LTE modem start using this mode for all clients, whether they're bridges or not, thus making it impossible for normal clients to connect. (I haven't actually seen this happen, but it's technically possible.)