Titleist ball experts recommend sticking with the same ball fit for you regardless of weather conditions but if you feel it works for you to switch out and that gives you confidence in your shots, by all means do what you think works best for you.

All balls lose distance as the temp drops. There’s no playing advantage to switching to a softer ball.
Not to say that a softer feel like the AVX over a Pro-V isn’t appreciated by our joints that are also a little stiffer as the thermometer goes down.
If you can adjust for flight window and rollout, then use the softer ball.

If the AVX feels better then use it
Both are good balls
Use what feels comfortable

Yes you are correct. The AVX feels a bit softer off the clubface. Ideally you want to play the same ball year-round, but I know what you're saying when playing in cold weather. If it feels good, do it.

All balls lose some durability (cold, wintery conditions are more likely to scuff ball) And cold air is denser than warm air causing you to lose a few yards of distance. So like Titleist experts say I stick with the same ball, because your next ball option will not get it's normal performance either.

Like most say use the ball you feel comfortable with. Things you might consider is the spin on the balls. The Prov1 has 388 dimples, Prov1x 348, AVX 348 and Prov1x - dash 328. Less dimples, less spin. Guess might be one reason Titleist does recommend using the same ball the year round. Ball reacts the same. Just use more club in the wind and on colder days. Go with your choice but look at how the balls will react. I am approaching that beautiful number, 77, play along the coast of SC and yes we do have some colder day, play a lot in the wind, and love my Prov1s the year round. Just love that some feel every time that ball leaves my T300s. We hit our drivers about the same.

I use the same ball regardless of weather and certainly don't find much difference when it is cooler

The was a test done by MyGolfSpy on the impact of cold temperatures on the golf ball. In there test, ball speed dropped by 6mph (unavoidable) but the driver spin increased by 500rpm (almost 20%). The launch was also a little lower. I tend to be a higher spin player so when I play in the cold (I play my share of round under 50F in Canada), I often switch out my ProV1X Left Dash for a AVX to further reduce the spin. The lower compression also feels better so it is an added bonus.

The AVX has a higher spinning dimple pattern 348 than the Pro1x left dash 328. The left dash has the lowest spinning dimple pattern of the premium balls. Your left dash is designed Players looking for high flight and low spin. Where the AVX is for players who prioritize long iron distance through low spin and soft feel. I tried the left dash prior to the new Prov1s coming out. Was giving some white boxes (prototypes) back in October and never looked back. Have always played the Prov1 year round, and the new ball outperformed the left in every aspect. Have also played the AVX. Staying with Titleist recommendation, consistency in your golf ball. Their R&D is second to none. Am a tester, member, and contributor to MyGolfSpy.

The AVX is a very popular winter ball at my club for most ProV1/X players.

Regardless what ball I play, I keep two in my pocket and rotate them from hole to hole. I figure keeping them as warm as I can can't hurt.