This seems like a lot of work to save ~$30 on a battery that's designed to do the task required.
An 18v 5a cordless battery has 90 watts of stored energy, a 12v 7A SLA battery has 84 watts of stored energy and
you can buy one of those for $24.99 delivered to your door and not have to fart around with odd power connections. Sometimes DIY is good, it can save money and achieve a better result. I don't think that applies here.