Matching existing texture is difficult. Most texture has several coats of paint that makes a smoother surface than the new, thus probably the reason for your "faint swirl". First will be to determine what was used to do the original swirl, possibly a sponge. Next will be trying to match what is there. Sometimes a good drywall person can come close to matching an existing texture but not exactly, so you may want to try doing just the patch and see what you think prior to doing the entire ceiling. Remember YOU will notice the patch more than others because you know it is there. Others may not ever see the imperfections. If you are not up to doing the task I would suggest having a professional come and give you some ideas on what is best. Depending on their level of skill they may want to blend it with the present texture, maybe not. As for cost, that is difficult to estimate without looking at the project. I kind of use a dollar a sq.ft for large areas which includes one coat of paint. That is my western Kansas price and is subject to change. I like to work by the hour as I believe that with most small jobs it is more fair to both parties. If I give you an estimate I am sure to cover any and all possibilities of things I belief I might get into with the repair. If you trust me to work by the hour, you pay me for work actually done. But remember the person you hire by the hour better know what they are doing and must preform in an efficient and professional manor. Get a few estimates, for a package job and by the hour. Talk with each and ask them to describe how they would repair the ceiling. Then go from there. This is probably to much information but it didn't cost much. Good luck.

You can drill the dent out with a hole saw or spade bit. And then glue in a round oak plug (1/4" thick). Sand, wire brush (manually), apply brown/smoke reactive stain and white hard wax oil.

When you say no air born, is that just fear of overspray? Or thinking the job is too small? The reason I ask is for a small repair I used a small aerosol can of finish. When I looked for the product I used, this helpful website popped up; maybe something you can use? https://www.thespruce.com/how-to-repair-orange-peel-texture-4153933

I've never used hot mud, but I'd apply the texture in the final skim coat (I'm very neat when I work w/ mud :) Practice creating the pattern on a separate piece first, so you know how to achieve the match. A brush or comb (something with teeth) is most commonly used to make that swirl pattern.