I think the trainer is probably giving you sound advice, though he may be overly cautious, he was there with you in person and had a lot more information available to base that decision on than us (and as a fitness professional is potentially highly educated, a lot of the personal trainers are pretty much just people who happen to be in good shape, and may not be all that qualified really... but there are also a lot of them that have so much physiology education they are half way to being doctors).
Now the good news, even without doing a lot of running your ability to run will greatly improve as you get to a good bodyweight for your build. I ran for over a year without much improvement, seems like I couldn't handle much more than a mile, there was a bit of gradual improvement, my main exercise was weightlifting but I always felt like the goal isn't just to look good, it's to actually be a functional athlete (which running is kind of a symbol of), well, that slow progression in running started to take off after a while, when I started getting close to my goal weight and somewhat lean (still overweight) but I found the difference between being able to run a fairly fast 1.5 miles and running 3 miles, was about 10 lbs for me. At 217 lbs I could do a respectable 1.5 miles (probably less than 10 mins) and at 207, 3 miles was surprisingly easy (mostly the fact that I could do 3 miles non-stop at my natural pace is surprising).
So my own advice would be stick to ellipticals and bikes, weights, and diet, a little running if it's comfortable is fine, but it's probably going to be better and easier overall to progress in running once your weight is more in line with your body type (which may or may not fall into the 'ideal weight' range of BMI or other measurements). Conditioning in running is something that builds up quickly and fades quickly, but your weight will drastically limit your endurance, your lungs don't grow from exercise, it's more about the rest of your body not having to consume as much oxygen to feed your muscles to keep up the activity.