DaVinci's Cat by Catherine Gilbert Murdock (2021) Greenwillow Books

DaVinci tried to create a machine to travel from one place to another using a cabinet.  His tests did not produce the results he expected, so he discarded the cabinet, making a gift of it to the Pope.  What he did make was a time machine.  But his cat entered the cabinet before it was shipped and came out as a kitten when a young man, a hostage, or a guest of the Pope on the condition that his father, leading the Pope's armies might not go over to the enemy, opened the cabinet.  He was delighted.  When the cat went back into the cabinet, it disappeared.  When it returned, a man came with it.  He was interested in getting his hands on discarded sketches from Michelangelo or Raphael.  One was of a girl no one recognized.  This, of course, upset the Pope that there was a thief running around in the Vatican.  This man returns several times, bringing the young man chocolates with peanuts.  But the last time he comes, he ventures out of the Vatican Palace, which he should not do, and finds a sick girl.  He runs back to the cabinet and disappears.  The next time the cabinet is used, a young girl comes out with the cat.  She is the girl in the sketch.  Only she wants Raphael to do the sketch and sign it so she can save the sick girl, who has grown old by now.