Should the GST cost be included when claiming rental management fees or investment management fees?

Yes, you can include the GST paid on expenses for rental expenses.

Thank you - and is the same true of Investment management fees?

yes, any expense. The only time you don't include the GST is when you are a self-employed business and you are a GST registrant and you are filing GST returns, then you have to exclude the GST from your business expenses.