Incremental Encoder and Arduino
This one is easy.  Even if you don't want to throw a microcontroller at this, I would most definitely NOT reinvent the wheel by making a hardware quad encoder system, especially as if you're designing for the UNO, you're probably not talking about production.

Use a purpose-built IC, like the HCTL-2022.  As you've realized, you've just hit a bugaboo having to do with how the output is latched for a read.  Before you're done, you'll hit three or four more, maybe having to do with something like noisy transistions.  These purpose-built ICs are pretty good.  I've used them and recommend them.