Would it be possible to get the X-chromosome and add SRY to create a "fuller" Y-chromosome?
Possible? Sure. You could sequence both chromosomes and then create some artificial DNA that included the combined sequences.
Would it be viable? Probably not.
What advantage does the skinny Y-chromosome give an individual anyway?
If there is a genetic condition in the father's X-chromosome, the male child is spared from it.
If there is a genetic condition in the mother's X-chromosome, the male child has a 50/50 shot at being spared from it.
If the genetic condition requires two homogenous X-chromosomes, the male child is spared.
If the genetic condition requires two heterozygous X-chromosomes, the male child is spared.
These are all advantages of simply requiring a Y-Chromosome, and there are certainly disadvantages to having a Y-Chromosome as well; X-linked genetic disorders will definitely affect any male heirs, unlike...
Females only use one X-chromosome (via the Barr body) so they don't seem to gain as much from having the pair.
...Female offspring. Because women have two X-chromosomes they have redundancy. If there is a bad gene on the extra X-chromosome that is deactivated, then it doesn't matter - where for a man with the same X-chromosome it would definitely affect them.
Also, because of the redundancy, what X-linked conditions do exist usually affect women less-so. If both X-chromosomes are utilized for a specific gene, then one bad protein product is partially made-up for with the one good protein product.