I have seen boats that have used polyurethane over the teak and the results are not very good in the long term. The polyurethanes tend to become cloudy over a long period of sun exposure; and re-coating with topcoats also results in heavy build up of finish that is cloudy because polyurethanes are not as clear as varnishes to begin with.
I have seen excellent results with Epifanes; but it is high maintenance as a low uv-resistant finish. The new Epifanes with UV blocking may be a better choice; but I have not used it and therefore can't compare it to a product like Cetol. All standard varnishes are higher maintenance than Cetol; you will only get appx 6-12 months of protection out of them before you need to scuff sand and re-coat to prevent UV degradation. If you don't re-coat on this schedule the sun will damage the finish and everything will need to be stripped off and re-finished.
I would not use Awlgrip over an existing varnish. Awlgrip and DuPont Imron (and possibly others) contains a cyanide based catalyst which is extremely harmful if inhaled. If inhaled while being sprayed it can be deadly. If you hand paint it on I would use a respirator that can trap cyanide vapors. In industrial application the people who spray this type of paint use a positive pressure type respirator to prevent any inhalation of the paint fumes.
Varnishes are safe to use; if you don't like the smell of the paint fumes you can use a respirator. If you are applying it to the interior you probably will want a respirator so you are not breathing too much of it. For exterior finish it's probably OK not to use one.
If you go with a varnish you might want to consider making sunbrella covers (or having them made by a canvas shop).