Below points worked for me while upgrade to Magento 2.4.4 EE
php bin/magento maintenance:enable
composer require magento/composer-root-update-plugin ~2.0 --no-update
take back up of composer.json
composer require magento/product-enterprise-edition 2.4.4 --no-update
Please run below commands after successful upgrade:
php bin/magento cache:clean
php bin/magento cache:flush
php bin/magento setup:upgrade
php bin/magento setup:static-content:deploy -f
php bin/magento maintenance:disable
setup:static-content:deploy -f command If you want to use
setup:di:compile command you can do it but It might be throw some errors we can do it later.
Just for cross checking version upgraded or not, upgrade and deploy command is enough.