You can try only adding the VLAN for management, something like :
/interface vlan
add comment=MGMT interface=ether1 name=VLAN200_MGMT vlan-id=200
/interface lte apn
set [ find default=yes ] apn=internet.xxxxxxxx.xx passthrough-interface=ether1 passthrough-mac=auto use-network-apn=no
/interface list member
add comment=defconf interface=ether1 list=LAN
add comment=defconf interface=lte1 list=WAN
add interface=VLAN200_MGMT list=LAN
/ip address
add address=192.168.188.1/24 comment=myconf interface=VLAN200_MGMT network=192.168.188.0
this way the pass-through would remain the normal ether1 (no VLAN) and you would only need to set the VLAN 200 for connection management.
Though still no idea if it would work and why the Winbox connection doesn't without this stuff.
Personally I would like better (if possible) the suggested "other" (that should be simpler) workaround, making use of a bridge, but I cannot find a valid example to try.
I am not sure how to exactly implement Sebastia's suggestion here: