Are we allowed to wear PTs while walking our dog?

Assuming your dog is wearing a reflector belt -

AR670-1
10–3. Occasions for wear
a. All-purpose wear. The physical fitness uniform is authorized for wear on and off duty, on and off the installation, unless restricted by the commander. Soldiers may wear all or part of the physical fitness uniform with civilian attire off the installation, unless restricted by the commander. The physical fitness uniform is not intended for wear as an all- purpose uniform when other uniforms are more appropriate.
b. Approved wear. The physical fitness uniform is prescribed for year-round wear for all Soldiers, unless otherwise directed by the commander. Soldiers may wear the physical fitness uniform off-post unless prohibited by the commander.
c. Restrictions on wear.
(1) Soldiers may not wear the physical fitness uniform for commercial travel.
(2) Personnel may not wear the physical fitness uniform in off-post establishments, unless for purchase of essential items (for example, gas).
(3) The physical fitness uniform is not considered appropriate for social or official functions off the installation, such
as memorial services, funerals, weddings, inaugurals, patriotic ceremonies, and similar functions. (4) Commanders may further restrict wear of the physical fitness uniform per paragraph 2–5c. (5) The physical fitness uniform is not appropriate for parades, reviews, and ceremonies.

Yes, walking your dog as PT with no other stops like the Shoppette could be considered an occasion for wear, especially if you are walking your dog before or right after organized PT. Sure I get it, but other wise, why would you?

Why would you want to?