A couple of questions since your header asked for dog friendly.

Do you want your dog to be able to roam over and possibly lay down in this ground cover?

Are you looking for something that is pollinator friendly, native or non native?

Is there water available, is the soil sandy, loamy or clay?

How large is this area and what do you want it to look like?

Usage and soil type are the 2 most important.

Correct, looking for something for dogs can roam over and lie down on. Something if they ingest that won’t harm them.
we don’t necessarily care whether it’s a pollinator, native . More important is that it can withstand dogs running on it

Mary is about 600 ft.² And is on mix of loam and sand. How it looks is not really important but rather  two have something covering the ground? 

Correct, looking for something for dogs can roam over and lie down on. Something if they ingest that won’t harm them.
we don’t necessarily care whether it’s a pollinator, native . More important is that it can withstand dogs running on it

Mary is about 600 ft.² And is on mix of loam and sand. How it looks is not really important but rather  two have something covering the ground? 

Have you considered grass?  Given the heat we have been having each summer a mix of tall fescue, fine fescue, a little perennial rye and a microclover seed kept at 4-6" would create a cool, soft place for them to lay and would also allow them to roam without creating bare spots. We have a trial patch of 4 seed mix tall fescue and microclover 'Pippolina' at the Extension office in Sussex County. It's in full sun and not irrigated, so part sun and some irrigation would have it looking a lot better.  You can come and view this plot.

There are grass mixes made for dog usage that are typically called Eco lawn mixes. They tend to include a large amount of fescue (all different types) which are hardy under wear and tear (and digging).  Specific dog lawns usually call for a large % of Kentucky Blue Grass which doesn't do very well in the southern DE heat and humidity unless it is in full shade.

If you don't mind brown grass from October - April you could plant a mix of warm season native grasses ( Buffalo grass, blue grama and side oats grama) which stay short and are green in the summer and brown out in cold weather.  These only need to be mowed 3-4 times a year.

In part shade, a Carex such as Carex pennsylvanica could be installed as plugs to create a soft, mounding ground cover.  This carex is semi evergreen and fairly tough once established. It spreads by rhizomes similar to grasses.  These could be inter-seeded with Violets, Viola sororia, a native groundcover that would not be harmful to the dogs and is  comfortable to walk on, extremely hardy, grows well in part shade and can fill in bare spots. It can spread easily by seed and is not readily dug up, i.e. a long tap root; which means the dogs will have a hard time rooting it out as well.