Welcome! Wow, it looks like you're well on your way already!
Usually if the stars look bloated in astrophotos, it's because they're overexposed. But 5 seconds, f/2.8 at ISO 1600...seems unlikely here. What stacking software did you use, and how did you align your individual frames? A little bit of misalignment can definitely make them look bigger. What are you using to process your images? There are a few techniques to reduce star sizes that are fairly specific to the software. For example, in Photoshop you can use the Noise | Dust & Scratches filter to essentially eliminate the stars from a (copy of!) the image. For an image like this the process would go something like:
1) Duplicate the background layer. Name the duplicate "starless".
2) Use the dust & scratches filter on the duplicate to get rid of the stars (apply as many times as needed).
3) Set the blending mode of this layer temporarily to Difference (i.e., with the original, untouched layer underneath). Suddenly it's nothing *but* stars! Create a clone layer of it (Mac: Command-Option-Shift-E; PC: Ctrl-Alt-Shift-E). Name this layer "stars". If you like, try using the "Minimum" filter to reduce the size of the stars.
4) Set the blending mode of "starless" back to Normal, and drag it to to top of the stack. Add adjustment layers however you like to stretch and enhance colors.
5) Once you're happy, drag the "stars" layer to the top of the stack (hiding "starless"). Give it a layer mask. Paint black on the layer mask with a soft brush to let the "starless" layer show through anywhere you want.
5) Creating a star layer - Imaging - Tips, Tricks and Techniques - Stargazers Lounge