Disclaimer: This answer is a work-in-progress, since final solution have not been found yet.

Roadmap overview

1. Clean-up the biber-specific features of your .bib file using biber-tool

2. ...

---

1/ Clean-up the biber-specific features of your .bib file using biber-tool

The .bib file might use biber-specific features (such as the @inbook entry, crossref or urldate fields, etc.). A new .bib file that is BibTeX-compatible must thus be generated. Following answer based on this and this treads.

• Create a file named myBiberConfig.conf containing following code. It defines how your original .bib file should be transformed in order to be BibTeX-compatible.

<?xml version="1.0" encoding="UTF-8"?>
<config>
    <output_fieldcase>lower</output_fieldcase>
    <output_resolve>1</output_resolve>
    <output_safechars>1</output_safechars>
    <sourcemap>
        <maps datatype="bibtex" map_overwrite="1">
            <map map_overwrite="1">
                <map_step map_field_source="date" map_match="[0-9]{4}?-([0-9]{2}?)" map_final="1"/>
                <map_step map_field_set="month" map_field_value="$1"/>
            </map>
            <map map_overwrite="1">
                <map_step map_field_source="date" map_match="([0-9]{4}?)" map_final="1"/>
                <map_step map_field_set="year" map_field_value="$1"/>
            </map>
        </maps>
        <maps datatype="bibtex">
            <map>
                <map_step map_type_source="conference" map_type_target="inproceedings"/>
                <map_step map_type_source="electronic" map_type_target="online"/>
                <map_step map_type_source="www" map_type_target="online"/>
            </map>
            <map>
                <map_step map_type_source="mastersthesis" map_type_target="thesis" map_final="1"/>
                <map_step map_field_set="type" map_field_value="mathesis"/>
            </map>
            <map>
                <map_step map_type_source="phdthesis" map_type_target="thesis" map_final="1"/>
                <map_step map_field_set="type" map_field_value="phdthesis"/>
            </map>
            <map>
                <map_step map_type_source="techreport" map_type_target="report" map_final="1"/>
                <map_step map_field_set="type" map_field_value="techreport"/>
            </map>
            <map>
                <map_step map_field_source="address" map_field_target="location"/>
                <map_step map_field_source="school" map_field_target="institution"/>
                <map_step map_field_source="annote" map_field_target="annotation"/>
                <map_step map_field_source="archiveprefix" map_field_target="eprinttype"/>
                <map_step map_field_source="journal" map_field_target="journaltitle"/>
                <map_step map_field_source="primaryclass" map_field_target="eprintclass"/>
                <map_step map_field_source="key" map_field_target="sortkey"/>
                <map_step map_field_source="pdf" map_field_target="file"/>
            </map>
        </maps>
    </sourcemap>
</config>

• Process your original .bib file with the --tool option of biber

  biber --tool --configfile=myBiberConfig.conf <yourbibfile>.bib
  

• Your processed .bib file is the freshly created <yourbibfile>_bibertool.bib

To add to the existing answers. For bibtex-2.16 a minimal configuration:

<?xml version="1.0" encoding="utf-8"?>
<!-- Got the date from https://gist.githubusercontent.com/mkouhia/f00fea7fc8d4effd9dfd/raw/500e9dbc6aa43a47e39c45ba230738ff4544709f/biblatex-to-bibtex.conf -->
<config>
  <output_fieldcase>lower</output_fieldcase>
  <output_resolve>1</output_resolve>
  <output_safechars>1</output_safechars>
  <output_format>bibtex</output_format>
  <sourcemap>
    <maps datatype="bibtex">
      <!-- Easy type conversions -->
      <map>
        <map_step map_type_source="report" map_type_target="techreport"/>
        <map_step map_type_source="online" map_type_target="misc"/>
      </map>
      <!-- Date to year, month -->
      <map>
        <map_step map_field_source="date"
        map_field_target="year" />
      </map>
      <map>
        <map_step map_field_source="year"
        map_match="(d{4}|d{2})-(d{1,2})-(d{1,2})"
        map_final="1" />
        <map_step map_field_source="year"
        map_match="(d{4}|d{2})-(d{1,2})-(d{1,2})"
        map_replace="$1" />
        <map_step map_field_set="month" map_origfieldval="1" />
        <map_step map_field_source="month"
        map_match="(d{4}|d{2})-(d{1,2})-(d{1,2})"
        map_replace="$2" />
      </map>
      <map>
        <map_step map_field_source="year"
        map_match="(d{4}|d{2})-(d{1,2})" map_final="1" />
        <map_step map_field_source="year"
        map_match="(d{4}|d{2})-(d{1,2})" map_replace="$1" />
        <map_step map_field_set="month" map_origfieldval="1" />
        <map_step map_field_source="month"
        map_match="(d{4}|d{2})-(d{1,2})" map_replace="$2" />
      </map>
    </maps>
  </sourcemap>
</config>

Coupled with a few command line options:

biber --tool --configfile=biberConf.xml references.bib --output-file refsTmp.bib --output-legacy-date --output-field-replace=location:address,journaltitle:journal

Is sufficient. More details are here. Note that sufficiency in this case is defined as being recognized as valid bibtex by sphinx, and this should not be used without more work for other cases (like journal templates).