If the holes are small enough and you don't care about spacing measurements, you could use what are called pin gauges. These will obviously not work on non-circular holes (irregular?)
These are available in sets of various ranges. The precision is quite good, often measured in tenths of a thousandths of an inch.
If that is excessive in terms of precision and especially in terms of cost, there's a much less elaborate method you can consider.
I initially planned to suggest modeling clay, but it has limits with respect to handling. You would flatten a sheet of the stuff and press it against the surface with sufficient force to create bumps as it enters the holes.
Once peeled off, you should be able to measure the bumps to get the hole sizes.
I think it would be difficult to handle, however, without breaking it.
As an alternative, especially as you have access to the inside/reverse, is to use a silicone molding compound of some high Shore value (hardness). The clay might be useful to create an inner dam as well as a container of sorts for the silicone on the outside.
Silicone molding material, especially the high hardness stuff, is quite durable. This should allow the portability and handling you require. Some compounds have minimal shrinkage, allowing for reasonable precision. It may be necessary to test sample holes to determine exact shrinkage levels, but it would be consistent across a specific range of diameters.
Expanding from the above suggestion would be to seal off the holes with adhesive tape rather than modeling clay or other dam material.
The ideal situation would be that your curved surface has a horizontal orientation, minimizing the damming requirement and the amount of silicone used.