How can I use influxdb_client_3 on AWS lambda?

I want to query my InfluxDB Serverless v3 instance from AWS lambda to retrieve some simple data.

I’ve now spent 8 hours or more trying different approaches but no luck so far.
Here’s what I tried:

I defined an AWS Lambda function with the Python 3.12 runtime and just put in
from influxdb_client_3 import InfluxDBClient3
This fails (of course):
"Unable to import module 'lambda_function': No module named 'influxdb_client_3'"

So I tried to add a layer to my Lamda that contains the package(s). I followed several guides you can find online that all are doing something like this (here on a Mac):

mkdir packages
cd packages

python3 -m venv venv
source venv/bin/activate

mkdir python
cd python

pip install influxdb_client_3 -t .

rm -rf *dist-info

cd ..
zip -r python
aws s3 cp s3://my-bucket-name

I then create and add this new layer, so I can import influxdb_client_3.

But this fails with the following error message:
"Unable to import module 'lambda_function': No module named 'pyarrow'"
I believe this is due to the fact that I packaged it on my Mac and Lambda needs the correct pyarrow library for the underlying Lambda architecture.

So my next try was to to the steps above on an AWS CloudShell (Amazon Linux 2023 x86_64), but this didn’t solve the issue either.

Can anyone provide a simple step-by-step guide on how I can just run a query on AWS Lambda against my InfluxDB Serverless instance?