How to add trusted root CA to Docker alpine

Suppose I am at network where there is MITM SSL swaping firewall (google.com is not issued by Google, but reissued by custom CA root authority) some more details here https://security.stackexchange.com/questions/107542/is-it-common-practice-for-companies-to-mitm-https-traffic .

I have simple Dockerfile:

FROM alpine:latest
RUN apk --no-cache add curl

It fails badly with error with SSL errors

 => ERROR [2/2] RUN apk --no-cache add curl                                                                                                                                    1.0s
------
 > [2/2] RUN apk --no-cache add curl:
#5 0.265 fetch https://dl-cdn.alpinelinux.org/alpine/v3.13/main/x86_64/APKINDEX.tar.gz
#5 0.647 140037857143624:error:1416F086:SSL routines:tls_process_server_certificate:certificate verify failed:ssl/statem/statem_clnt.c:1913:
#5 0.649 WARNING: Ignoring https://dl-cdn.alpinelinux.org/alpine/v3.13/main: Permission denied
#5 0.649 fetch https://dl-cdn.alpinelinux.org/alpine/v3.13/community/x86_64/APKINDEX.tar.gz
#5 0.938 140037857143624:error:1416F086:SSL routines:tls_process_server_certificate:certificate verify failed:ssl/statem/statem_clnt.c:1913:
#5 0.940 WARNING: Ignoring https://dl-cdn.alpinelinux.org/alpine/v3.13/community: Permission denied
#5 0.941 ERROR: unable to select packages:
#5 0.942   curl (no such package):
#5 0.942     required by: world[curl]
------
executor failed running [/bin/sh -c apk --no-cache add curl]: exit code: 1

Every tutorial at Internet says that I can add own "trusted" root certificate and run update-ca-certificates.

But it can be added by "apt add" only. This situation seems to me as "chicken-egg" problem.

FROM alpine:latest
USER root
RUN apk --no-cache add ca-certificates \
  && update-ca-certificates

Error is similar

=> ERROR [2/2] RUN apk --no-cache add ca-certificates   && update-ca-certificates                                                                                             1.0s
------
 > [2/2] RUN apk --no-cache add ca-certificates   && update-ca-certificates:
#5 0.269 fetch https://dl-cdn.alpinelinux.org/alpine/v3.13/main/x86_64/APKINDEX.tar.gz
#5 0.662 140490932583240:error:1416F086:SSL routines:tls_process_server_certificate:certificate verify failed:ssl/statem/statem_clnt.c:1913:
#5 0.663 fetch https://dl-cdn.alpinelinux.org/alpine/v3.13/community/x86_64/APKINDEX.tar.gz
#5 0.663 WARNING: Ignoring https://dl-cdn.alpinelinux.org/alpine/v3.13/main: Permission denied
#5 0.929 140490932583240:error:1416F086:SSL routines:tls_process_server_certificate:certificate verify failed:ssl/statem/statem_clnt.c:1913:
#5 0.931 WARNING: Ignoring https://dl-cdn.alpinelinux.org/alpine/v3.13/community: Permission denied
#5 0.932 ERROR: unable to select packages:
#5 0.933   ca-certificates (no such package):
#5 0.933     required by: world[ca-certificates]
------
executor failed running [/bin/sh -c apk --no-cache add ca-certificates   && update-ca-certificates]: exit code: 1

Is there some other solution how to install update-ca-certificates tool? Or am I missing something?

Thx

See @kthompso answer for working solution.

Working solution (with update-ca-certificates commnad) based on @kthompso answer and info from unable to add certificates to alpine linux container

FROM alpine:latest

USER root

# To be able to download `ca-certificates` with `apk add` command
COPY my-root-ca.crt /root/my-root-ca.crt
RUN cat /root/my-root-ca.crt >> /etc/ssl/certs/ca-certificates.crt

# Add again root CA with `update-ca-certificates` tool
RUN apk --no-cache add ca-certificates \
    && rm -rf /var/cache/apk/*
COPY my-root-ca.crt /usr/local/share/ca-certificates
RUN update-ca-certificates

RUN apk --no-cache add curl

Edit: One solution I have in my mind is to use curl docker image with -k option and download .apk with those certificates and tools. Install it as local file. Add my root CA certificate and run update-ca-certificates. It sounds super crazy, so I think that have to be better solution :)