[discodermolide]

I think you might need to move towards a more "modern" method of alkyl-alkyl coupling involving a boronate or a triflate (from the alcohol), halide and a transition metal, Pd is usually used, but I think Cu and Ni can be persuaded to do the job too.

[Enthalpy]

Thanks Discodermolide!

I'll try to find literature and see if I grasp something...

[Enthalpy]

Appended here is a reactor example that separates mono-, di-... halocyclobutanes as they form. Unless I'm mistaken of course.

It's straightforward and usable for other compounds, so it must exist already. Books must refer to lab experiments, not to a continuous process, when they tell that the degree of halogenation is difficult to control.

Here X₂ could also be a hypochlorite, a mix of halogens, or H₂O₂ in case it works
http://www.chemicalforums.com/index.php?topic=80085.msg296423#msg296423
A slow influx of halogen or equivalent may help control the quickest ones, a bit of added Br₂ or I₂ too. Some halogens need cooling rather than permanent heating. Not represented neither is the evacuation of HX.

The distillation can also use a packed column. To harvest only one degree of halogenation, a single reactor that condenses that one suffices. Two or more reactors permit to scoop both the mono- and dihalo- here, where independent pressures let adjust the temperature. Separated reactors can also use different halogens.

1atm boiling points in °C estimated by Mpbpvp:
  Cl    Br    OH
 +13   +13   +13  C₄H₈ cyclobutane
 +81  +108  +112  C₄H₇X
+125  +174  +182  C₄H₆X₂
+159  +232  +242  C₄H₅X₃
Well separated.

Marc Schaefer, aka Enthalpy

[Enthalpy]

I've stumbled over the very text telling why dicyclobutyl is easy to produce. In:
 Novel Metathesis Chemistry, edited by Y. Imamoglu, L. Bencze
page 336 and around (excerpts on Google Books).

Metathesis of easily obtained methylenecyclobutane (optionally with dimethylenecyclobutane), followed by hydrogenation, give dicyclobutyl (and tricyclobutyl), as sketched here. The desirable mix of both is supposedly even easier to get. Heavier compounds don't hurt in small amount and are easily separated.

Caution: the performance table in the book isn't credible. Maybe their "kerosene" reference is very bad, but dicyclobutyl gains only 4s over the usual RG-1, and the spiro compounds are denser and more efficient than the cyclobutyls. Biased comparison.

For safety (and density), tricyclobutyl is better, or at least a mix. Then, I prefer a 2+2 photocycloaddition of vinylcyclobutane as on August 08, 2015 if it works. If correct, the data for linear tricyclobutyl is already very pleasant: bp+190°C, mp-47°C, 890kg/m³, just half a second behind dicyclobutyl.