Try mintty[1], comes default with Cygwin these days. Not perfect, but a lot better than the alternatives I've tried. It supports select-to-copy and middle-mouse-to-paste, which is something that's very deeply embedded in my muscle memory.
Select-to-copy is yuck for two reasons: 1: After you select, it copies and then deselects. This is annoying if you want to see that you selected the right thing, and it doesn't allow you to change your selection if you see you were off by a character. 2: I like to select to mark my place sometimes, but I don't want that to change what's on my system clipboard.
ctrl/shift Insert is yuck because it's a two-hand keystroke and doesn't have an easy way to remember whether shift or ctrl is the copy. (I don't often make mistakes between the two, but it leaves me unsure of myself which is a half second lost to hesitation each time.)
I miss middle-click to paste and select-to-copy where ever I can't use them.