What you are initially proposing, to take a spur off the ring final to the outbuilding would be OK if it was
limited to 13A. So a FCU (fused connection unit) is added to the ring and that feeds your 2.5mm cable and then in the out-building you can have sockets and another FCU with 3A fuse to switch/protect the lights circuit.
But if you need more than 13A, which if you want a 3kW heater and a fridge kicking in at 5A+ every so often it looks like it, then you need really to go with #2 post by
@JBW175 suggestion and get in a dedicated new supply. If that is too expensive, then look at limiting your heater to 1-2kW max.
Speak to a local electrician (one who is registered with one of the competent person schemes like NICEIC/SELECT/etc) about it, they are probably happy for you to put in the armoured cable so long as they type/supplier is one they know and trust.
2.5mm is really only OK for 20A or so which should be fine here, but is too much to be taken off a single point of the ring final. But for all the extra cost of the cable, if it is going in as a dedicated feed then you might want to get it sized for more so you can have stuff in the future like a big hot-tub, or car charger, etc, depending on the garden layout, etc.
Also you really want them separate from your house sockets, either dedicated feed or a double-pole isolating FCU, so in the event of a fault you can isolate the out building and restore your home electrics until such point when it is fixed. As it will most likely be middle winter and pouring down rain, etc at the time!