I'd tell him. When he brings it up, tell him you appreciate his concern about you, and how he does not want to burden you, but you love him and would not find taking care of his dog a burden.
Make it all about you, not about OW at all. And be totally non-accusing. You know well inside how to tell him so that he'll listen.
Sorry, I had not been about the bb much lately, I've had a few rough days...
"You don't throw a whole life away just 'cause it's banged up a little"
Tom Smith in "Seabiscuit"