Yes, it is almost certainly possible to artificially synthesize just about any protein. There can be difficulties with certain proteins, but these are more engineering problems rather than true limits to artificial synthesis.
The problem is getting the protein where it needs to be to be functional!
A great comparison is insulin. Insulin is an important regulatory protein that floats around in the bloodstream. That makes insulin therapy a relatively straightforward solution to a problem of not enough insulin: all you have to do is make or isolate some insulin and inject it into the bloodstream.
Dystrophin, however, is a structural protein that acts inside cells. Proteins are really big, and they can't be simply taken up by cells. There's no mechanism to get an arbitrary protein inside: any proteins that can be taken up by cells need to do so via some specialized mechanism.
If you wanted to replace a defective allele for dystrophin, then, the most feasible approach would be gene therapy: getting a patient's own cells to produce a functional copy of the protein. Efforts to entirely or partially replace dystrophin via gene therapy are a current area of research, but there are still many challenges to gene therapy approaches in general.
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