Bringing in a bike from overseas to south-east Asia shouldnt be too hard, but of course a bit costly.
The problems start when you want to cross borders. Singapore and Malaysia shouldnt be too hard if you have a Carnet and insurance.
Thailand introduced new laws and regulations in 2016 for foreign vehicles making it mandatory with excessive permits and mandatory guides to enter the kingdom and travel within it. However as of lately it seems from the many reports I have been reading on different travellers forums that it is relatively easy to get into Thailand without the mandatory permits and guides through most bordercrossings except the Mae Sot - Myawaddy bordercrossings between Thailand and Myanmar. They seem to still enforce the rules with mandatory permits and guides.
So upon entering Thailand without guides and permits one should be given the 30 day TVIP as was the norm before. But for you this will cause a problem - how will you enter Thailand, find a place to safely park and store youre bike, fly back home, do your things back home, fly back to Thailand, pick up your bike and ride it across a border within those 30 days TVIP allowance you got when you entered the country?
That said - I have heard of travellers who have gotten both 60 and 90 days on their TVIP, but that seem to be more luck than visdom...
Leaving a bike for 4 weeks - I guess would be easiest in Malaysia or Cambodia. In Malaysia on a Carnet and in Cambodia some borders dont issue any TVIP or other documents so I guess one can have the bike there as long as one want(?)
Thailand would be problematic more than 30 days, the same with Laos. Myanmar would be impossible.
Laos and Cambodia do also to some extend make it troublesome to enter their countries on a foreign bike - but it should be ok if one do the homework before one try to enter these countries.
Myanmar - mandatory guides and permits are required! No way around that for the moment. Best to do it in a group to get the costs low. I have seen 7-10 day crossings as low as 6-700 USD including guides and permits and accomodation. So it isnt that extremely costly but still a bit...
Bhutan - I have travelled Bhutan on a bike but that was on a arranged group tour. Beautiful country indeed - but as far as I remember you can only ride in Bhutan through an pre arranged group tour and guides were mandatory back then. Its possible this have changed by now as the tour I did was some 7-8 years back in time. But I doubt this ad I havent heard of many single riders travelling in Bhutan.
Another option is to buy a bike in Thailand or Malaysia. Its possible, in Thailand it just takes a sertificate of recidency and some time and the bike is yours legally. And with thai plates it could be stored as long as you want. You should relatively easily be able to travel Malaysia, Thailand, Laos and Cambodia on it. If you get a Carnet for it you can also do Myanmar, India, Nepal etc. Maybe all the way to your home country...?