Before you get too deep into issues about cultivation in a shaded area, ask yourself how much you actually like Tayberries. After all, you already have two bushes. Would you not prefer something else for variety?
Personally I am not a big fan. At their best, yes they are very good, but I have always found they go from being under ripe to being over ripe very rapidly. So getting them in perfect condition is tricky. But they are liked by many and some on this site are really enthusiastic fans.
Blackberries have a reputation for being the best fruit for partial shade. Worth a thought.