No The Doctor clone does not have the sufficient items to grown his own TARDIS...
The reason is the following...
As stated in the Wiki for TARDIS
TARDISes are bioships that are grown from a species of coral
presumably indigenous to Gallifrey, as stated in "The Impossible
Planet" (2006), and it can take years to complete one.
The main thing to take from this quote is that the species of coral that TARDISes are grown from are most likely only indigenous to Gallifrey as stated.
The problem is that the scene where The Doctor gives the coral to the clone is a deleted scene... Now then deleted scenes are not considered canon, so we can ignore that fact. After taking consideration that we have not seen any actual coral in a live broadcast of Doctor Who, perhaps it may have been shown Pre-Time War but it's not shown in New Who, we can conclude that there are no more in existence because Gallifrey is gone... Perhaps it's really rare, and it's not stated if the coral is grown inside The TARDIS either... So the clone of The Doctor would not have ever gotten his hands on one.