Once you migrate the instance to Azure SQL Managed Instance the Edition name is "Azure SQL Managed Instance" as mentioned here. The Id of the edition is 8 as shown on the query below:
SELECT CASE ServerProperty('EngineEdition')
WHEN 1 THEN 'Personal'
WHEN 2 THEN 'Standard'
WHEN 3 THEN 'Enterprise'
WHEN 4 THEN 'Express'
WHEN 5 THEN 'SQL Database'
WHEN 6 THEN 'Azure Synapse Analytics'
WHEN 8 THEN 'Azure SQL Managed Instance'
WHEN 9 THEN 'Azure SQL Edge'
WHEN 11 THEN 'Azure Synapse serverless SQL pool'