Ritt BIB is one of the more corny/bourbony ryes out there... If you are getting *too much* mint on that one, then I'm not sure where to go.
As noted the LDI Ryes are going to be like gargling mouthwash. FWIW: I find the Russel's Reserve Rye to be quite minty/piney as well.
As suggested Wild Turkey 101 is a good bet, as is the Baby Saz - but fundamentally the very nature of Rye means it will have a fruity/spicy character to it - so what you pick up as mint is supposed to be there in some amount - the higher the rye, the mintier it might be - unless well aged.
I get almost no mint on Vintage 21 & VWFRR 13 - more juicy fruit and Christmas spices, but Saz 18 has a fair amount of mint.
The Thomas Handy Rye is also not minty - it's an explosion of flavor.
But for the price point of a Ritt, none of those are close, and most are terribly hard to find at any price.
Baby Saz
Wild Turkey 101 (if you can still find it...)
Beam or Old Overholt (which I don't like)
Those are the only "barely legals" out there in that under $30 price point that won't be mint-bombs. *I* would also include Ritt BIB on that list, but obviously our pallets don't agree!
FWIW: My brother despises mint, but Baby Saz is his favorite whiskey... so, go figure.
As noted above... you may want to stick with Bourbon.