Hey
miquelj, you have a decent profile and a pretty good GMAT score. You are on the right track with your target schools, Target schools as well, they need to be a bit streamlined though.
Frankly, I don't see your profile as a challenge but I am a bit dicey about how you are going to portray it in your application. This is because I feel that you might be under-estimating your extra-curricular and work experience, a mistake that most applicants make. When you really brainstorm on your profile, even seemingly ordinary experiences can be portrayed in an extra-ordinary manner which might just be the differentiator you need to land an acceptance letter. In the same context, you have mentioned that your grades were affected due to some personal challenges - You can use your essays to justify the showcase your story and justify the reasons for low grades.
You have not mentioned your college rank - Tier-1 schools like HEC are notorious for recruiting primarily from top ranked schools so if your college ranks within the top 25, it will be a leverage.
Secondly, if you want an In-depth analysis of your profile strengths and weaknesses, you can take up a free profile evaluation here:
https://mim-essay.com/profile-evaluation?gcfWith a strong application, I would say that LSE can be an option and can be bucketed under a dream school.
If you have 6 months or so in hand, you can take up a few more profile building activities - working on a CFA certification, taking up another quality internship, joining an NGO etc.
Hope this helped, good luck!