How to get costs for Specific Group within Bridge Model Design

In your pbix, Fact_Costs[CostAmount] was of text data type, so I used SELECTEDVALUE as that didn't require any data transformation or doing something like SUMX/VALUE. Really, you can use any function or measure where I used SELECTEDVALUE, which is the direction you are already going in your question. E.g.

SUM if Fact_Costs[CostAmount] is a number format:

Costs_Number by Group = CALCULATE( SUM( Fact_Costs[CostAmount_Number] ), Bridge_ResourceGrouping )

Same output but leaving column as text:

Costs_Text by Group = 
CALCULATE(
	SUMX(
		Fact_Costs,
		VALUE(Fact_Costs[CostAmount_Text])
	),
	Bridge_ResourceGrouping
)

Or, here is a calc for resource average, to get away from very simple SUM examples:

Resource Cost Avg = 
CALCULATE(
	DIVIDE(
		SUM(Fact_Costs[CostAmount_Number]),
		COUNTROWS(Dim_Resources)
	),
	Bridge_ResourceGrouping
)

Table of above measures with Dim_Group[Group Name]:

added resource count to help validate the avg measure