No. We do not record your text over a protected melody, the rights belong to someone else and a licence costs many times the price of the song. You get an original melody written for your words instead, with a chorus the room can join by the second time around.
The short answer is no, and the reason is not convenience. A well-known melody belongs to whoever wrote it, and the recording belongs to whoever made it. A licence for that has to be cleared per country and costs many times what a whole song costs here. Anyone promising you a chart melody for the price of a gift is promising something they cannot deliver.
What you get instead is a melody written for your words, in the style you pick: pop, ballad, rock, schlager, folk. The advantage is bigger than it sounds. With a borrowed melody every syllable has to be squeezed into an existing metre, and that is exactly where most home-made birthday songs fall apart. Here the music follows your lines rather than the other way round.
If the goal is a room singing along, a new melody does that too. The chorus repeats by design, which is the part a group can join after the second pass without knowing the words. Choose the choir voice, print out the lyrics, which you get in full, and play the song once before the moment itself so people recognise the tune.
The other tradition, new words written onto a melody everybody already knows and sung by the guests themselves, is a different craft and needs no recording at all. It works best if you do it yourself: take a melody the whole table can hum, then write your lines syllable by syllable into that same metre and hand out the sheet.
So there are two routes to the same evening: a finished recording with an original melody here, or a rewritten text on a familiar tune that you sing yourselves. Plenty of people do both, the recording during dinner and the printed rewrite after dessert.
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