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I don't understand why you think you need to get close.
In general, the longer the focal lens, the less distortion you will get. That is why portrait lenses are longer than standard. As you don't want much DOF you would be even better of with a moderate telephoto.
A potential problem is that modern consumer lenses have some distortion uncorrected in the knowledge that the camera will correct them. Such a lens would have to be used with a suitable camera.
My starting point wouild be a Tamron SP 90mm macro, which covers a field 10cm wide x 7.5cm high at a distance of about 50cm (m4/3 x2 crop factor).
Close working distance options:
The SP 17mm, with 7mm extension will cover 5cm wide with 5cm working distance. It is a rectilinear lens.
The Olympus OM 24mm 2.8, with 7mm extension, will cover about 4.5cm to 7 wide with 5 to 7cm working distance. It looks very sharp, with the meerest hint of barrel distortion at the very edges of the frame.
All of the above are manual focus film lenses.
I hope some of this helps.