Use pool name as variable from variable group for deployment task

This question came from fact that you can only use global variables for deployment: https://docs.microsoft.com/en-us/azure/devops/pipelines/process/deployment-jobs?view=azure-devops#schema

Use only global level variables for defining a pool name. Stage/job level variables are not supported to define pool name.

But my config is a bit more complex. I do have main pipeline.yml with several stages. Stage's jobs defined in the template, so I have following structure:

# pipeline.yml (some cuts)
....
stages:
  - stage: Build
    displayName: Build all
    jobs:
     - template: ../templates/build.yml
       parameters:
        stage: 'DEV'
  - stage: Dev
    displayName: Deploy Dev
    variables:
      - group: GROUP-DEV
    pool: 
      name: '$(env._global.agentPool)' # env._global.agentPool defined in in the variable group
    jobs:
      - template: ../templates/deployment.yml
        parameters:
          stage: 'DEV'
   # other stages here ....

# templates/deployment.yml
parameters:
    - name: stage
      displayName: 'Stage'
      type: string
jobs:  
  - deployment: Deploy${{ parameters.stage }}
    displayName: Deploy ${{ parameters.stage }}
    environment: ${{ parameters.stage }}
    
    strategy:
      runOnce:
        deploy:
          steps:
            - download: current
              artifact: art_${{ parameters.stage }}
              displayName: 'Download art_${{ parameters.stage }}'
            # more steps here.....

Now the problem: with such config I got error:

##[error]Pipeline does not have permissions to use the referenced pool(s) . For authorization details, refer to https://aka.ms/yamlauthz.

which means we faced with limitation mentioned in the first quote (proof: https://developercommunity.visualstudio.com/t/pool-name-in-deployment-job-cannot-be-in-a-variabl/940174)

As soon as I change either pool name to hardcoded value or remove -deployment and use regular steps all works fine.

Question: with that, is there any chance to re-work templates to still use agent pool name from group variable? I need the -deployment task to reference environment.

What was tried already:

• use pool name as param of the templates/deployment.yml and use pool inside steps. No luck, same story