Your assumption is correct, as your son needs to use an amount of his tuition that exceeds $5,000, he has Zero left to transfer.
And the answer to your question is "No, he is unable to transfer an amount that has not been reported on his own return". Unfortunately, CRA requires that the taxpayer must claim their T2202 - Tuition & Education Amounts in the year it is incurred. This means if he was not to report it on his own return, he could not transfer any amount to you.
TurboTax will apply any needed tuition amounts to reduce the tax payable to zero this year and carry forward any excess to future years.