My hypothesis is that The Royal Mint discovered that it had a problem manufacturing proof dies for this new 1½ pence coin design using conventional methods.  Previous coin dies, like the Maundy Coins, had proof like fields that could be easily polished.  The Mint apparently opted to issue matte proof process instead, which was a method previously used by France.  

Proof dies were different than normal dies for two reasons. Although the dies used were standard dies, the Mint selected better than average dies with a lot of detail.  Plus, the proof dies used more pressure than a normal business strike would use.  Also, the matte proof dies were not used for as many strikes as a normal business strike die. This resulted in strongly defined strikes with strong lettering and high quality rims and dentils.

Proofs were officially minted in 1837, 1862 & 1870.  It is believed that the 1834, 1838 & 1843 dates were struck unoficially.  At least one is known to exist of each of the above dates.  The unrecorded 1834 seems to be the rarest and most valuable proof date.

Sometimes, It's kind of hard to differentiate between a proof and a standard business strike.  In order to be sure that a 1½ pence coin is a proof, the coin must show certain point of “diagnostics”, that are small features in the dies that will transfer to all coins struck by those dies. Furthermore, each of the proof coins that I have been able to study properly, have strong details, and a thick and sharp edges, opposed to the thin edges of business strikes, although this could differ from specimen to specimen. It is believed that these proof examples were removed from there original packagings early and not so well preserved, creating over the time an wide range of rainbow colors or interesting toning.  

Collectors must be wary of cleaned, whizzed or altered examples. Thus It is advised that collectors stick to purchasing certified examples of proof.  If you are buying an uncertified example, make sure you have sufficient knowledge to make the diagnostic and can spot the differences in proofs versus normal business strikes.     

Δ  Comparatively in these two photos show one of the diagnostic point, perhaps the most important according to my hypothesis. The first picture display a proof circulation strike example. Market with an arrow the edge diameter, which range from 0.82 to 0.84 mm. The bottom photo shows a normal circulation strike coin, also market with an arrow, indicating the edge diameter which range from 0.77 to 0.78 mm.

Proof mintage was made in small batches dependent upon perhaps, private requested orders. As no records were kept of the mintage, is commonly assumed that approximately 5-10 pieces of each three halfpence of the dates [1834, 1837, 1838, 1843, 1862 & 1870] that are known or are thought to exist were struck in proof. To this day I have not been able to locate any image or provenance information pertinent to the date 1843.

During these early years an appropriate pair of dies, sometimes new but often used, was polished and used to strike the required number of coins. Aftermath some of the proof dies were usually returned to general use, where they would result in proof-like coins until the polished surfaces wore down.

This is why some proof strikes may show the same details of a variety among the circulation strikes. The 1870 was possibly struck as a souvenir of this denomination, since 1862, the last time it was minted for circulation. There have been examples sold at auction as Proofs but these were actually proof-like business strikes.

Is been said the Mint, had them shipped as gifts for some important authorities. Overall, all these factors, limited production, coupled with the fact that recipients of the coins, had little or no knowledge of the importance of them at the time, storing them improperly [this assumption may be drastically wrong], etc. would mark the Proof three halfpence coins very rare [R7] to extremely rare [R8] in all grades.