BH says poor presentation can have an element of inheritance, but whilst I've been here we haven't had the same cows do it more than once. So personally I wouldn't let them stop me keeping a ewe lamb on.
On going straight back in after a breech or otherwise held up / prolonged birth, especially where there are or might be triplets - you won't do any lasting damage to the ewe (but of course give her antiobiotics if you've been in and vitamins after a difficult or assisted birth) and if you delay you may have a dead lamb. So yes, in general I'd always rather let the ewe do things at her own pace but in such a case I'd put the lamb or lambs first.
With an experienced ewe, especially something roomy like a mule, and very hormonal and maternal, I wouldn't worry about the intervention causing any issues. Sometimes with other, less maternal and/or less roomy sheep you may get some resistance to the lamb if she's had a difficult birth - but they usually do come round after it's had a few feeds.
We do get loose skin on some of our Texel lambs; BH always says they are the lambs that turn out the best. Can't see the skin under the macs in your pics, so can't say whether these look loose in the same way.
Was the Bowmont tup raddled? If so, you'd expect him to mark any ewes he tupped on the first day - but some other colours won't show over red, of course. (Always start with the lighter colours
. And don't use blue if you are also using yellow then green
)