I have no experience with this..but, I have worked with an acrylic supplier that could bond acrylic together and it was completely invisible when done. So I know what you propose can be done. Acrylic can be machined, sanded, and polished to look water clear, so repairs can be whittled into something you can live with. Although, a bad repair could be worse that the original problem or so I reckon.
A quick google brings up these solvent based bonding agents which are probably the way to go if chosen and used right. Shooting from the hip here a bit, sorry.