I am not aware of any pen kits which have the "barrel push" integrated with the click mechanism.
The current "rage" in kit pens with click mechanisms are these :
1. . The Duraclick EDC pen kit from PSI - uses the German Schmidt SKM-88 click mechanism
2. . The Luxor pen kit from William Wood-Write - uses the German Schmidt SKM-88 click mechanism
3. . The Sierra Super Button Click pen kit from Berea Hardwoods - uses the German Schmidt SKM-88 click mechanism
My preference is for #2, but I won't tell you why unless you write to me and ask .... the reason may not be what you might expect.
There are other kits that use the small Schmidt SKM-192 click mechanism (very similar to the SKM-88 in operation but very different in appearance).
Frankly, I do not recommend these unless you want to make a slimline clicker - just as quiet but more awkward to use.
Of the pen kits listed above, the PSI Duraclick EDC was the first to appear, right around the early part of 2019, so that's only 2 years ago.
Happy to offer more info if you want it. . I have tried all of the above kits, and most of the others using the same or similar mechanisms.