The caption of an enumeration value is the text that the end user sees for this enumeration value, So you just have to change the caption of it and the value will remain the same.
If Asha's suggestion is not sufficient, you could create a second enumeration and some logic. This would be an option e.g. if you want to retrain historic data of records that should not be updated anymore.
Your original BE_ENUM would have values enum1, enum2, enum3, enumeration1
The second one (BE2_ENUM) would have enumeration1, enum2, enum3.
For read only records, you could show BE_ENUM, for records that you need to update, you could show BE2_ENUM. Of course, you will have to create some logic as well.
1) A microflow that you run after deployment of this change to set the values for the attribute of BE2_ENUM, based on their current value in BE_ENUM (you will have to decide what to do with the enum1 values!!)
2) An onchange microflow on the attribute using BE2_ENUM that will set the corresponding value in the attribute linked to BE_ENUM.
That way you can still use other logic related to BE_ENUM (of course needs to be updated slightly based on your new enum value)
This is a high level example and could be different based on your use case. If you expect more changes in the future, I would suggest to refactor your solution so it can use configurable values that the admin will manage. You should then add an 'outdated' flag that you can use to filter only your current set of values in dropdowns.