I think one should always train for max strength, but I think that your max strength won't be what it would be if you were only lifting and also eating a lot.
I'm in this Conditioning forum because I also enjoy running and walking and hiking, so I'm willing to give up some strength (though I think the worries about losing strength and mass are over-stated while doing cardio) in return for my conditioning.
The thing is I think if you drop weight that will help a lot for that one-arm pull up. I notice that when I'm at my heaviest (270 about 7 or 8 months ago) then I can only do about 3 pull ups in a set, with only about 4 or 5 sets possible in a day.
Now at 245 I can do 5 sets of 5.