You can go in relative safety by road on the Chandigarh-Kullu-Keylong-Leh route. It does not traverse any of the troubled parts. It is a seasonal road that will open first week June and close by mid-September (give or take a week). Very scenic, but will test your biking skills.

Regarding bike rentals; one-way rentals are uncommon and you might end up having to pay full cost of the bike as a refundable deposit. You can hire a bike in Leh for travelling in the surrounding areas - in my experience, most of these bikes are in disgraceful condition.

A German friend bought a good condition second-hand Royal Enfield in Delhi, used it for 2 months and sold it back to the dealer for a net loss of INR 8000 (this was back in 2012). If you know someone in India who you can trust, and can provide you with the necessary paperwork - this would be the way to go.