This in not an authoritive answer - but my best guess.
I understand the Personal Ordinariate of Our Lady of Walsingham is not open to let Latin Rite catholics change rites to Anglican Rite Catholic. You must convert to this from Anglicanism (or related sect) or be born into it or otherwise be baptised into it.
The same will be true of the equivalent Anglican Rite Catholic churches in other countries.
However Latin Rite Catholics are welcome to participate in Anglican Right Catholic Liturgies.
that said. Evensong is derived from the Divine Office. This has a current Latin Rite version, and it is fully appropriate for a Latin Rite parish to publicly celebrate this.
It is commonly known as “Vespers” or “Evening Prayer” this latter name is where the term “Evensong” originate (assuming a sung celebration of this prayer).
Within Latin Catholicism, the public celebration of the Divine Office is part of the “Cathedral Tradition” and part of the tradition of all Religious orders (though some have their own variations)
The General Instruction to the Divine office reccomends that the Divine Office is used as the structure or at least the inspiration for most prayer groups or public worship outside the Mass or other defined structures.
It is Fully and Highly appropriate for you to ask to form or participate in Evening Prayer in your Parish Church. If you or others you can encourage to assist have the musical skill this can and if possible should incorporate music. this may be from any appropriate tradition.
However the version and translation of the Divine Office used should ideally be from the officaial Divine Office approved for your country or region.
Using other versions may be permissible, but it is then considered a personal devotion, not a formal liturgy (this is similar status to the Rosary which is a personal devotion not a Liturgy). When celebrated in accordance with the approved texts, it may qualify as a formal Liturgy.
Your Priest or Deacon should support you in this and will hopefully be willing to participate and lead it. However this Liturgy does not require an ordained man to lead it if one is not available.
There is no problem with drawing inspiration from the Anglican rite musical forms and using them in a Latin Rite celebration of Vespers.
Link to the approved text on Amazon.com
Morning and evening Prayer
Most Catholic bookshops will stock this or other version and all can get it.
It’s a sub-set of the full Divine Office with is a 4 volume set published by St. Josephs Press in the US or a 3 volume set published by Collins in the UK. - Officially you must use your countries version as the translation varies and so do the saints days and some other feasts.