'having been' in past perfect

I am trying to understand how to move the expression having been better to past perfect case in the following sentence:

They consider some period in the past having been better than the times we live at.

Something like this (although sounds odd)?

They considered some period in the past having had been better ...

The verb consider is a catenative verb. Catenative verbs take a nonfinite construction as their complement. This nonfinite verb can be in participial or to-infinitive form - or both as in the case of consider.

• You should consider watching less Netflix. (gerund-participial)
• I consider Netflix to be better than Disney Plus. (to-infinitive - present)
• I consider Netflix to have deteriorated recently. (to-infinitive - perfect)

In the participial construction consider means to think about. In the infinitive construction it means to be of the opinion that. Note that the infinitive construction has to be preceded an object:

I consider [noun phrase] to [verb phrase]... .

Your sentence They consider some period in the past having been better than the times we live at uses consider in the sense of to think, to be of the opinion that. So neither of the participial constructions you suggest is possible. (And indeed the having had been + past participle construction does not exist in English.)

Amending your sentence to use the perfect infinitive results in:

They consider some period in the past to have been better than the times we live at.

A final point. You can even omit the perfect infinitive in this context. For example:

They consider the Middle Ages better than the times we live in now.