Recognize that you are asking for travel conditions normally associated with high end travel.

-small number of travellers (private tour)
-ability to dictate own itinerary

Myanmar is far from fully independent travel. It's only fully been open to travel groups for a few years.

I would recommend reaching out to high end hotels and arranging a guide and car for the day if you really want to get off the beaten track. The Governors Residence in Yangon can arrange this.

Alternately, you probably need to find a group to go with up country to Bagan / Mandalay that caters to budget travelling.

I can't help you out with the budget travel. Other than the backpacker culture hasn't fully arrived just yet.