[syNRG]

I have the HM5 with the E11 and yes you can get some good bass on them. 

 

[syNRG]

If I can't get at least similar bass to the m50's when the HM5's have a $50 amp, then can you recommend a headset with similar quality and sibilance, as well as bass, while having a better soundstage? Please remember though, they need to have minimal leakage to the outside, and be able to be worn for very lon extended periods of time, like around the 10+ hour mark.  
Also please, remember that I really don't want to go that much higher than around $150-160, as when I've asked before, people keep on recommending setups that are around $200.

Ok reading this and another comment which said that the M50 bass can get a bit out of hand/uncontrollable at times makes me think the M50 with the E11 (I also read that you didn't want the E11 since you can't use while charging but i'm going to go ahead anyways as i'm sure you will pick up an amp with a bass boost feature). The HM5's I feel do get out of hand when you use the bass boost feature of the E11 when used on the higher of the 2 settings.
The HM5 do not leak to bad in my experience and are extremely comfortable. I do believe that the M50 suffer from a bit of both with the stock pads, which is why a lot of people have opted to get the Shure SRH840 pads which use memory foam to replace the pads on the M50. So you could also get the M50 with the Shure pads to compliment them. This choice is up to you of course. I almost know for sure that the HM5 is more comfortable no matter what you do that the M50 since they just feel so light and almost weightless, but the clamp is a tiny too much out of the box. 10 hours seems a bit excessive but the HM5 is exteremtly comfortable, if not the most comfortable headphone under $200. 
Good luck and remember that I haven't tried the M50 headphones, but I do own the ATH-Pro700MKII (since I didn't want a more neutral headphone like the M50 since i own the HM5, I wanted more bassy headphones which is why I got the pro700).
Hope you find what you want

 

[SoundPon3]

Here's an idea......
What about getting the jaycar rebadged HM5s (cheaper price for the same thing, which is always good) and a FiiO E17k? The E17 can be a USB DAC/AMP combo, standalone amp and standalone DAC so you can use it with your phone and bass shouldn't be an issue with the EQ settings on the E17. Also have a look at the E07k (or E7k?). It's pretty similar but I don't think it has as many features.
Cheers,
Sound

 

[Dehaha1]

The FiiO E17 by itself almost maxes out my budget, so I don't think that will be a possibility.

 

[syNRG]

  The FiiO E17 by itself almost maxes out my budget, so I don't think that will be a possibility.

 
I would say that even though the E11 cannot be used while charged, I would go with the E11. It has a long battery life, longer than any smartphone. I think I read somewhere 20+ hours of use. You should just charge overnight and use it the next day