Performance Issues on MSSQL Server Web Edition
These types of decisions should be backed up by a lot more investigation and testing. However, from the information you have provided I can give you a few pointers to put you in the right direction to make the right decision. 


Since you are already spiking CPU to 100%, reducing cores to save cost and get more memory will only make things worse. 
Also, check what is causing CPU spikes on the server, maybe some performance tuning for the code can help the situation. 
Why add 128GB RAM when SQL Server can only make use of 64GB on the server? Do you have any other applications on the server? If you do, move them to a separate machine, it will help with the CPU spike issue. 
Even if you do plan to upgrade to Standard Edition why add 256GB RAM, your SQL Server will only be able to use 128GB RAM. 
Collect some query stats on the server during peak hours to check if more memory will actually help the situation. 
Also, look into moving "old" data into a separate environment (data purging), to keep the application DB size small. (not an easy thing to do but is something should be on your todo list). 
Setup a test environment, to test your conclusion drawn from the above investigation to find out whether your suggested fix will actually work. Also, make sure you test your new solution in the proper test environment (Like prod) and not with 10 rows in the testing environment. 


Once you have done all the above, the chances are you will make the right decision or what ever you do will not be very far off the right solution. 

Also, if possible convince your business to buy the enterprise license, present them the case, educate them and let them make the decision.