Hello,
As I see your website is not big, there is not many product etc...I would suggest to create completely new website, but with same menu links, users, products (because of SEO) + additional stuff.
SEO: I will do your website up to standards (seo friendly links, meta data, inner links etc) all other aspects for SEO, like link building, article writing are not included into price.
Implementing payment processor is easy and semi-automated.
I would suggest one of the Gavick templates, they are stable, responsive and I have account with them, so you do not have to pay for it. Also they have implemented Shopping cart that is tested and stable.
Regards,
Milos