Try this approach, which is how it's done in factories.
Add some air to the tube until it's a sausage, but no more than needed to hold the shape. Stuff the tube evenly into the tire.
Starting at the valve, mount one side and use both hands to work to the far end in both directions. Flip the wheel, and mount second side, working both directions toward the valve. If necessary, go back and massage the tire toward the valve to get some slack and push it over.
Finally, push the valve in to the tire to make sure you haven't trapped tube under the bead, then pull back out.
Pump to 20psi or so, then spin to make sure it's seated evenly. Pump to riding pressure.
Note, the job should take less time than I spent typing this. Women in Taiwan do about 2 per minute.
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