No, you can overdose with certain multivitamins so this is not something i would recommend. What are you using for a multivitamin? D3 on its own is not a multivitamin so he is correct there. Where are your temps now at basking and what is the doc suggesting bumping it up to.
Multivitamins with D3 and or A should only be given 2 times a month due to both D3 and A being fat soluble so they store in the body and tissues. Edema along with toxicity issues can happen when there is too much given of either of these.
Keep an eye on it and on his behavior. You can see edema when something in the body is off. Make sure you are not seeing any other signs of an RI as well. The start of an RI is not going to be as obvious as a progressed RI.