For winter I went with General Altimax Arctics 205/65/16.
For summer I got Wild Country (generic label from VIP) 215/70/16.
Neither rubs and the speedometer difference is miniscule.
Those Generals are like velcro in the snow. I live in NH and travel throughout NH and Maine on all kinds of roads. Frankly, unless you're into off-roading, there is no need for 4wd if your road agent does the roads every few hours. 3-4" is not even close to a problem for a FWD Patriot with good snow tires.
The Wild Country tires have had 2 replaced for defects, and one for a legitimate flat tire (screw in tread too close to sidewall to be patched) -- 3 outta 4! I'm practically on new rubber a year later! However, I wouldn't mind if they wore out. They're noisy on the road and not very good in snow.
IMHO the stock Goodyear LS2s weren't all that bad -- I read a lot of cynical reviews, but in my experience they were certainly a very acceptable tire. They were a good highway tire, well above average in the rain, and they worked well enough in snow. I used them the first year I had my Patriot and never had a problem. Even took them off-road (in a situation I didn't intend to be in -- mud, ice snow, you name it) and they did their part to get me out of a pickle. Wore out kinda quick, though.
As for the TPMS sensors, no you don't need them; we all got by for a century without them. Just check them once in a while. However, in my state the vehicle won't pass inspection with an idiot light on, so you'll have to get inspected with the wheels that have the TPMS sensors installed.