I have a simple implementation of a delta-V calculator for bielliptic transfers here if you want a quick look at the underlying math: https://github.com/daeken/OrbitalRingsBook/blob/main/space-t... -- the DeltaVForBiEllipticTransfer function. (The calculator widget in the book doesn't use it, but I was primarily using it for getting high orbits in an effort to minimize plane change delta-V for another calculation and this code inherited it. Also, you can actually play with the widget in action here, though again it's only using Hohmann transfers due to the low fuel requirements in play: https://ors.daeken.dev/space-travel.html)

Happy to explain the math in more detail, but the code is -- I think -- pretty self-explanatory.

Edit to add: this function doesn't calculate the initial lift to the higher orbit; that'd be the difference between the speed of your circular orbit and the perigee speed at the higher orbit.

There wouldn't be much to add. In a one-mass system, the speed of a satellite at any point is a function solely of the radial distance r: v^2 = 2/r - 1/a. The semimajor axis a characterizes the orbit; is positive for an elliptical orbit; and is the the arithmetic mean of the closest and farthest points on that ellipse—i.e. the periapsis and apoapsis.

If you're at one of either the closest- or farthest- points, at distance r, and the opposite one is s, this reduces to

    (defun speed-at (r s)
      (sqrt
       (- (/ 2.0 r)
          (/ 2.0 (+ r s)))))

And; if you have an orbit burn at one of these points that's co-linear with your direction of motion—something that simply expands or contracts the periapsis or apopasis, as is the question here—then velocity addition is a one-dimensional scalar sum. Concretely: if you're in an orbit defined by the peri- and apo- distances (r,s), and you want an orbital maneuver, at r, that either increases or decreases the opposite distance from s to s*, the Δv you need is

    (defun Δv (r s s*)
      (abs (- (speed-at r s)
              (speed-at r s*))))

That's all the math needed for calculating delta-v's for the maneuvuers in the OP post. I.e., the Hohmann transfer between two circular orbits, of radius r1 and r2, is

    (defun Δv-hohmann (r1 r2)
      (+ (Δv r1 r1 r2)    ;; at r1, raise apoapsis r1 -> r2
         (Δv r2 r1 r2)))  ;; at r2, raise periapsis r1 -> r2

And the bielliptic transfer is a sequence of three of these. I think it's

    (defun Δv-bielliptic (r1 r2 r-inter)
      (+ (Δv r1 r1 r-inter)
         (Δv r-inter r1 r2)
         (Δv r2 r-inter r2)))

(I've left out the dimension-ful scaling factors; if you want a concrete calculation for a real object, you can scale everything throughout (inside the roots) by G*M, which is i.e. 398600 [km^3/s^2] for Earth).