The fly in the ointment here is the need for Expanding ammunition. Muntjac are covered by the Deer Acts so anyone using a pistol-calibre carbine needs their FAC Conditioned for HP or SP, and an expanding ammo holding. Range ammo (such as Lead 157gr FP) isn't 'designed to expand'.
The ban on rounds of 'expanding' pistol ammunition AND missiles was the real reason for the EU Directive on Firearms, which was duly enacted in the UK Firearms Acts (Amendment) Regulations 1992. This led to the Firearms (Amendment) Act 1997 whereby, due to an 'administrative oversight', ALL Expanding including SP rifle ammunition & missiles was prohibited. We're still lumbered with the consequences of the HS Michael Howard's incompetence.
The upshoot is that Hollow-Point and SP pistol ammunition & bullets are now expressly banned under international law. There isn't a quarry use for such under UK law, or an exemption as there is for the rifle calibres used in deerstalking.
That's a statement of fact. Even if an FLD grants the 'expanding' authorisation, a supply of suitable ammo & bullets for Muntjac/CWD stalking isn't going to be on every RFD's shelves. Just thought I'd mention this.....