Kindly

Thank you kindly, sir she said,

Kindly has many uses. It can sound like a hidden reproof, but it can also be meant and understood literally, as in "She asked me kindly if I needed any help?"

Is that a split infinitive Oreo? I'm thrilled if it is, I've never spotted one before!

Shouldn’t that be ‘Could I ask you to kindly not let your dogs do etc’ ?

Kindly unhitch your wagon from that star, buddy!
Ogden Nash 😁

Ive been mulling this over, and have to say that I dont have an issue with someone kindly asking /asking kindly for something

In actual speech, as opposed to the written words, we would usually put on a kindly or maybe even a humble voice if asking a favour, wouldnt we?
I just see it as a written version of that.

(That said, "Could I kindly ask you not to let your dogs do their business on my front lawn" would definitely not carry that friendly tone, whether in writing or speech!) 😃

I am totally in agreement with you regarding the wording of the invitation, though.

Dunno. No one uses it down here. Sounds very formal.