Apply different bibliography style to specific printbibliography in biblatex

In general it is not possible to use different styles for different priontbibliography calls in a document, but very often it is possible to emulate the behaviour.

The following solution emulates the reading list by copying all the relevant definitions from reading.bbx. Then it is just a question of when these definitions are executed, because some settings are used by both bibliographies. Luckily, here all problematic definitions could be delayed until the last minute and there are no clashes. This may not always be possible and in those cases we may have to rename macros or work around this differently.

documentclass[british]{article}
usepackage[T1]{fontenc}
usepackage{babel}
usepackage{csquotes}

usepackage[backend=biber, style=alphabetic]{biblatex}

makeatletter
InitializeBibliographyStyle{%
  globalundefbbx@lasthash}

defbibenvironment{readingbib}
  {mkreadingbib
   list
     {}
     {setlength{leftmargin}{0pt}%
      setlength{itemindent}{0pt}%
      setlength{itemsep}{bibitemsep}%
      setlength{parsep}{bibparsep}}}
  {endlist}
  {bbx@item}
  
defbbx@item@false{%
  itemsep2bibitemsep
  itemrelax
  itemsepbibitemsep}

defbbx@item@true{%
  bbx@item@full}

defbbx@item@full{%
  itemsep2bibitemsep
  @itempenaltyz@
  itemrelax
  begingroup
  samepagebfseries
  deffinentrypunct{strut}%
  usebibmacro{entryhead:full}%
  ifbool{bbx:entrykey}
    {defnewblockpunct{%
       nobreakhskipz@skipstrut
       hfillpenalty100hskip1emrelax
       hbox{}nobreakhfillstrut}%
     deffinentrypunct{%
       parfillskipz@finalhyphendemeritsz@
       parnobreak}%
     newblock
     printfield{entrykey}}
    {}%
  finentry
  endgroup
  hrule height 1.25ptrelax
  itemsepbibitemsep
  @itempenalty@M
  itemstrut
  @itempenaltyz@}

defbbx@item@name{%
  iffieldequals{fullhash}{bbx@lasthash}
    {bbx@item@false}
    {itemsep2bibitemsep
     @itempenaltyz@
     itemrelax
     begingroup
     samepagebfseries
     deffinentrypunct{strut}%
     usebibmacro{entryhead:name}%
     finentry
     endgroup
     hrule height 1.25ptrelax
     itemsepbibitemsep
     @itempenalty@M
     itemstrut
     @itempenaltyz@}}

newbool{bbx:entrykey}
newbool{bbx:annotation}
newbool{bbx:abstract}
newbool{bbx:library}
newbool{bbx:file}

DeclareBiblatexOption{global,type,entry}[boolean]{entryhead}[true]{%
  ifcsdef{bbx@item@#1}
    {letcsbbx@item{bbx@item@#1}}
    {PackageError{biblatex}
       {Invalid option 'header=#1'}
       {Valid values: header=true, false, full, name.}}}
DeclareBiblatexOption{global,type,entry}[boolean]{entrykey}[true]{%
  setbool{bbx:entrykey}{#1}}
DeclareBiblatexOption{global,type,entry}[boolean]{annotation}[true]{%
  setbool{bbx:annotation}{#1}}
DeclareBiblatexOption{global,type,entry}[boolean]{abstract}[true]{%
  setbool{bbx:abstract}{#1}}
DeclareBiblatexOption{global,type,entry}[boolean]{library}[true]{%
  setbool{bbx:library}{#1}}
DeclareBiblatexOption{global,type,entry}[boolean]{file}[true]{%
  setbool{bbx:file}{#1}}

ExecuteBibliographyOptions{loadfiles,entryhead,entrykey,annotation,abstract,library,file}

newbibmacro*{entryhead:full}{%
  printnames[labelname][-1]{labelname}%
  setunit*{addcolonspace}%
  printfield{labeltitle}}

newbibmacro*{entryhead:name}{%
  ifnameundef{labelname}
    {printfield{labeltitle}}
    {printnames[labelname]{labelname}}%
  savefield{fullhash}{bbx@lasthash}}

newbibmacro*{entrytail}{%
  newunitnewblock
  begingroup
  defnewblockpunct{item}%
  ifbool{bbx:annotation}
    {usebibmacro{annotation}%
     newunitnewblock}
    {}%
  ifbool{bbx:abstract}
    {usebibmacro{abstract}%
     newunitnewblock}
    {}%
  ifbool{bbx:file}
    {printfield{file}%
     newunitnewblock}
    {}%
  ifbool{bbx:library}
    {printfield{library}%
     newunitnewblock}
    {}%
  endgroup}

newcommand*{mkreadingbib}{%
  DeclareNameAlias{author}{default}%
  DeclareNameAlias{editor}{default}%
  DeclareNameAlias{translator}{default}%
  %
  DeclareNameWrapperAlias{author}{default}%
  DeclareNameWrapperAlias{editor}{default}%
  DeclareNameWrapperAlias{translator}{default}%
  %
  DeclareFieldFormat{entrykey}{mbox{bfseries##1}}%
  DeclareFieldFormat{annotation}{bibstring{annotation}addcolonspace ##1}%
  DeclareFieldFormat{abstract}{bibstring{abstract}addcolonspace ##1}%
  DeclareFieldFormat{library}{bibstring{library}addcolonspace ##1}%
  DeclareFieldFormat{file}{bibstring{file}addcolonspace url{##1}}%
  DeclareFieldFormat{shorthandwidth}{##1}%
  %
  setlength{bibitemsep}{0.5baselineskip}%
  setlength{bibparsep}{0pt}%
  %
  renewbibmacro*{finentry}{%
    ifbibliography
      {usebibmacro{entrytail}}
      {}%
    finentry}}
makeatother


begin{filecontents}{jobname.bib}
@book{elk,
  author    = {Anne Elk},
  title     = {A Theory on Brontosauruses},
  year      = {1972},
  publisher = {Monthy & Co.},
  location  = {London},
  keywords  = {recommended},
}
end{filecontents}
addbibresource{jobname.bib}
addbibresource{biblatex-examples.bib}

begin{document}
Lorem autocite{sigfridsson}
nocite{elk}

printbibliography[title={Citations},notkeyword=recommended]
printbibliography[title={Further reading},keyword=recommended,env=readingbib]
end{document}