If I start the year with an amount in one category in the budget and I decide that I wish to move this amount to another category during the course of the year, is it possible to record this as a transaction? Or is the only way to do it to edit the amounts in the budget to reflect the change from one category to another?

Budgets won't create transactions. So, just edit the budget. Or, if you want, copy your existing budget and rename it to "Original budger", then make the desired changes to your budget.