I've got similar issues proportion-wise (long legs but short arms instead of bad back) which led me to get a custom frame made for most of my riding. As mentioned above, the stem might be OK with drops or other bars but you need a longer reach for flat bars, hence Cotic offering shorter top-tube versions of their Roadrat frame for use with drops. Try a Nitto technomic or similar stem with a decent rise/reach but you might want to experiment with some cheap old mtb quill stems (loads on ebay usually) before spending that money.
I've got similar issues proportion-wise (long legs but short arms instead of bad back) which led me to get a custom frame made for most of my riding. As mentioned above, the stem might be OK with drops or other bars but you need a longer reach for flat bars, hence Cotic offering shorter top-tube versions of their Roadrat frame for use with drops. Try a Nitto technomic or similar stem with a decent rise/reach but you might want to experiment with some cheap old mtb quill stems (loads on ebay usually) before spending that money.