Sounds perfect to me voltage wise mate, if it supplies 14+ volts while under full load it should be OK. You could still get a current test (Amps) done at somewhere like Halfords for piece of mind?
I would think that the problem is the battery. If it has a dead cell then the alternator will be trying to charge a battery that will only ever hold 10.5 volts at best, and on full load while your driving it the alternator is basically running everything on it's own.
If you can afford it, then I can't recommend :
Bosch Silver Battery HSB012 highly enough. If you find someone close to you with a trade card it's about £60, £103 is OTT. 5 year warranty as well
Other reliable brands are Exide or Varta