Hi,
I rode Manali - Leh solo and self-supported in August, 2017. My experience was slightly different. Having one's own bike saves a lot of money, considering you will take 7 days to go up to Leh and 2 days to drive down to Manali on a bus. Moreover, good condition bikes were over 750-1000 INR per day in 2017 when I inquired. May be that was rate for single person. Group might be less. If you have your cycle, go with it! And also, you wouldn't have to worry about mounting rental if you take extra days.
Front suspension is of course good to have. Hybrid will be fine for the route. My friend who did it a few weeks earlier than we was on a tourer and did fine. Over 80% of the route is good tarmac now.
As for terrain, except for other side of Rohtang Pass (after the climb from Marhi) the entire route has good surface/tarmac. The next bad stretch is a couple of kilometers after Baralach La pass. So, I feel literally any cycle can do, as long as the rider is not reckless.
August (first week) felt a bit late in the season. It rained all the way from Manali to Sarchu and got difficult to ride. Moreover, the passes snowed in and had high wind gusts. So, please keep very good quality gloves and raingear, if you want to be there in August.
Best
Sachin