I'm a firm believer of the "Keep It Simple Stupid KISS" brigade and my initial thoughts on your proposal look OK. I am not familiar with your version of the BIP ( mine has allowance for a battery) but I can't see why your proposal wouldn't work at this moment. By connecting your battery to the output of the transformer, you are presenting the BIP with a ready made, smooth 12Vdc supply. I believe the output of these earlier BIPs is not a great 12Vdc supply, somewhat lumpy perhaps but sufficient for inacandescent lights but not, maybe, for charging phones etc.. So, by sticking 12Vdc into the rectification circuit, you will most certainly get a smooth 12Vdc output. Perhaps a purist would connect the 12Vdc battery supply to the output stage of the BIP thus negating the BIP circuitry all together other than the distribution to the various 230Vac and 12Vdc circuits.
In my own case , I have replaced the battery charging circuit with a mains powered Ctek charger. At all times I have kept the original components intact should I need to retract but so far I have found very simple ways to improve the electrics in my Eriba without too much fuss or upheaval.
KISS