My suggestion is to not switch to a flat handlebar, but something like a Velo-Orange Porter bar. The 23.8 size will accept the brifters. The handlebar provides a lot of hand positions, and easy hand movement and access to the levers. It has a 15mm rise or drop.
My experience was that a flat bar did not work well on a road bike, too limiting on positioning, leading to discomfort. Adding in the cost of having to go to flat bar shift and brake levers is likely a factor.

If cost is not an issue, then there many options for swept back and riser type handlebars. Most of them have 22.2 grip area diameter. Some of which will fit bar end levers, most will not. Soma Oxford bar is one that is 22.2, but the inside diameter fits bar ends.

Both V-O and Soma have other options worth looking at.
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