It would be more believable if she is told that the first baby died, but was so badly malformed that she couldn't be allowed to see the body. A common belief at the time was that a children took on the appearance of things the mother saw or thought about. She may be upset by the decision, but she would accept that they were protecting the twin, and her own reputation, by keeping the body from her.
Otherwise, if he had the midwife in on the scam, and you established that she was a sheltered and gullible person, it is possible they could gaslight her in to believing there was only one baby. But it would be a hard sell.