Here's my thoughts from the last 9 months of overlanding in the sib-continent;

Re the carnet: I've no experience of extending a carnet in India, but have exported a vehicle (a truck) in and out of India several times over the last 6 months. I'm not aware of any rule regarding how long your vehicle must be out of India, and have had no problem at Customs re-entering with the vehicle after only a week out of India (via Nepal). Indian land borders are generally very easy in my experience; don't get me started on the monstrous hassle of importing a vehicle by sea port.

However, immigration officials are quite strict at enforcing the rule that you must have been out of India for 60 days before re-entering, even if you have been given a visa to do so. It makes a nonsense of multiple entry visas. Some immigration guards are bribeable on this, but not all. Indian consular officials in both Nepal and Bangladesh (and probably everywhere else, but these are the only places I have experience of) will NOT process an application for another Indian visa unless you have been out of India for 60 days, nor if you have many Indian visas in your passports. Having 2 passports doesn't get round it, as you can't swap passports at the border (the Indian immigration officials I've come across always look for your exit stamp from the last country).

Sometimes it's possible to get a 're-entry permit' from the Indian embassy in Kathmandu (and elsewhere, I'm sure, but NOT Dhaka) if you have more entries on your visa but have been out for less than 60 days, but it's hard and I wouldn't count on it. It'll last for max 10 days, and you must have evidence of flights into and out of India, though this can be easily forged, and you can then travel in and out overland.