I'll share with you my personal experience.
When I was a "just out of school" engineer, my first job was as a junior engineer in a construction company. My first assignment was as assistant of the field superintendent for the erection of a small sized fossil fuel power plant. The two boilers were Riley Stoker brand, made in the USA, working at 600 psig.
When we arrived at the jobsite, the civil work had already been done by a civil contractor and approved by the client. The two boilers' anchor bolts (a total of 24) were already embedded into the concrete foundations.
When we erected n° 1 boiler first supporting column, oh surprise, we found out exactly what you describe on your posting: the anchor bolts projection was insufficient and the nuts could not fully engage on them.
We thoroughly measured the projection of all of the anchor bolts and found out that the problem was going to repeat with all of them.
What to do? Blame the civil contractor and the client because he approved his job? Useless, of course. Break down the concrete foundations and remake them correctly? Impossible.
We explained the problem to Riley Stoker's field engineer who was there to supervise our job. He gave a telephone call to Riley's headquarters in Worcester, MA (in those days Internet didn't exist), and the answer was this:
1st. Put the nuts and tighten them up with an impact wrench as far as they will go.
2nd. Fill the hole left between the nut and the top of the anchor bolts with weld.
This was in 1962. More than 40 years have passed since then. I don't know right now, but two years ago the boilers were still there producing steam.
Giovanni S. Crisi
Sao Paulo - Brazil

There was a topic on this forum a few months back regarding welding of structural bolts. A lot of people were against the idea for various reasons and they were technically correct for the most part.
I will say that being in the structural steel business for quite a while, I have seen many nuts and bolts that had been welded. Similar to Giovanni's post, I also have not seen problems in those cases.

An A563 nut is very weldable. You will lose the benefits the heat treatment gives to the nut, but I would screw the nut on as far as you can and plug weld the recess with E7018. I would never weld a nut or bolt in a pre-tensioned or slip critical application but a column base plate as you described is not normally one of those applications. If loading is strictly in compression, those nuts won't be doing much now that the column is framed in. If your conditions are more complicated than that, then you might want an engineering review.

Just a side thought: I always find it interesting that problems with "short" anchor bolts always surface after the steel erection is beyond the point-of-no-return.

Another side thought: sometimes you can buy special nuts that will fit inside the base plate hole and extend downward to engage the threads.

CHGuilford