Tool/shears for cutting the backbone out of turkey
The brute force aproach


Typically when shears don't cut it, you bring in a saw:  

Butchers use them to halve cows or pigs, cut through large bones and generally chop down an animal in short time. For the home cook, the larger versions (bandsaw, hand-held circular saw, ...) are certainly overkill, but smaller ones are quite useful.

A hacksaw is in every butchers' toolbox (figuratively speaking) and available at every Home Depot or similar. Ask your shop assistant for the recommended sawblade and other details. The ones sold as tool might not be fully compliant with local food safety regulations, but if you don't plan to sell your food, a good scrubbing to remove oils and other residue from fabrication should suffice. 

If you plan to do some butchering - at least for small-to-medium sized animals - on a regular basis or love electric gadgets and have a few extra dollars, I found a small hand-held, cordless electric bone saw plus a video showing it in action. As I see it, it should be possible to operate it even with limited strength and dexterity.
 (Source)




The surgical aproach


My first impulse when reading your question was to approach it as a surgical problem - if your existing tools can't cut through the bone, you can always use a scalpel or similar knife to dislodge the ribs from their joints one by one. You don't get a nice clean cut, but can always trim a bit afterwards. This procedure shouldn't require much strength, a tremor might be a problem, though.




The completely different aproach


You might not need to remove the backbone at all, unless you insist on a traditional spatchcooked bird. the kitchn features an article on how to spatchcook a turkey "the Latin American way". 

In short, you bend the bird's legs outwards, slice towards the hip joint and dislodge the tighs. Cut towards the wings until you can flip the breast part up, then finally cut through the backbone: 



(More details and step-by-step instructions here.)

This means you are only dealing with some thin lower ribs and one cut through the backbone. You get two pieces, the breast (plus wings) with the whiter meat and the legs with the darker meat. It should be easy to procede from there.