Looking for Wide training shoes for backyard sprints (SIT)(HIIT)

I’m looking for shoe recommendations for workouts in my backyard.

Details:

• Size: 13

• Width: 2E my guess is have wide feet but not very wide to where size twelve is where my toes end in shoes but there to small so i have to go 13 for width

• Weight: 300

• Budget: under $200

What I’m doing:

• Short sprints back and forth (HIIT)

Surface:

• Mostly flat grass and dirt

• Sometimes wooden shed floor

What I need:

• Wide or wide-friendly fit

• Enough cushioning for my weight

• Stable enough for pushups and quick direction changes

• Good traction on grass/dirt

Not sure if I should get a running shoe, cross-trainer, or trail shoe.

What would you recommend