I could definitely hear the song, especially on the chorus.

This poem was a creative use of the prompt, applying the Lovesick Blues to a couple of dogs. In my head, I imagine it with a twang and and extended howling vowel on "blues" - something like "bloo-oo-oos". Sorry, but I have to make the tune up in my head.

The rhymes are all good and flow well. The little paws at the end is a nice touch

Suggestions: There are two lines where I felt the meter was lost and so a bit awkward. They don't need changing so much as rearranging, that is, just changing word order changes where the stressed syllables are and changes the rhythm. For example:

"Given was the account" sounds better to me as "Was the given account"

"And the payment was due" I might try something like "Payment coming due"

Now this is just my opinion which may be judged to be faulty. But I do the same on my own poetry when I read it aloud, sometimes the rhythm seems not quite there. I find a slight rearrangment of words often makes it sound more rhythmic.

Overall Impression: I found this poem/song to be a creative use of the prompt and very amusing. A subject matter most dog owners will relate to. This was a fun read, well done.