Let's be clear about the economics here.
The average cost per kWh in the UK at the moment is around £0.14.
A 110Ah 12V battery stores (very approximately) 110x12/1000 = 1.3kWh or £1.85 worth of electricity. It's not exactly that figure because the voltage is not 12V over the full discharge, and the 110Ah is quoted at the 20-hour rate but will be less if discharged in less than a day. Let's assume a reasonably optimistic 80Ah representing 100% depth-of-discharge to 10.8V over a full life of about 400 cycles. We can't really push it any harder because of the fairly aggressive charging rate required quite apart from the discharge.
The OP must supply his own battery, even if he's stealing the fuel used to generate the power. To get the performance mentioned he will need a good quality battery. 110Ah non-traction AGMs range from say £120-200. If he spends £175 he will get a good brand that will hold up to this kind of use. A traction battery rated for discharge at C/5 will do better at correspondingly higher cost.
Each daily cycle saves him 80 x 11.5 x 0.14 = £ 1.29 in electricity but costs 175/400 = £0.43 in battery wear and tear. So the maximum possible saving on this scheme is around 86p per day. I would be surprised if, in real-world changing conditions, you would achieve half that.
To quote the wise words of Dave OCD's post:
Why even bother.
FWIW, you can use a pair of diodes to allow a backup mains-driven PSU to take over seamlessly when the battery gets low. You set the voltage of the PSU equal to the minimum voltage to which you want to discharge the battery. Whichever is higher does the work. Use big, low voltage Schottkys to avoid undue losses.
As mentioned above, heavy cables will also be needed to avoid yet more undue losses. For those unfamiliar with transmitting power at 12V, the power loss is 367 times greater than at 230V for a given power and cable size (230/12)². E.g. the headlamp feeds on my boat are 10mm² (tunnel lights) and 16mm² (spot lights). 24V makes a 75% saving in copper over 12V.