Identifying unique values within a row whilst suming a column array

I am looking to sum a column of values, summing only values that are
1) is a "Weight" value
2) is greater than 0.001
3) is a unique value in its row

I can achieve the first and second points using SUMIF() and working with arrays (which I have recently learned about), but need help with #3. I am currently summing the data by hand, but as the data grows in rows and columns, I would prefer a more automated method, and one less prone to human error. I'm not sure there is another way to approach this problem other than a cell function?

Attached is a sample of the data. You can see the correct sums in "True total weights". I would like to achieve the same result in the yellow highlighted cells. Currently, values less than 0.001 are expressed in red fonts, and values that are duplicates in its row are highlighted pink.

Try

=SUMPRODUCT(--($A1:$A20="Weight"),--(B1:B20>0.001),--(COUNTIF(B1:B20,B1:B20)=1),B1:B20)

Not quite. I think the COUNTIF(B1:B20,B1:B20)=1 portion is checking for unique values within the column. I am looking to sum the column, but only values that are unique for its row in the table.

Is that more clearly phrased? I'm sorry if that was unclear before.

Also, if I may ask, what does the -- before each array in SUMPRODUCT() for?

Ahh, I'm not sure if I have a solution off-hand. It would involve a complicated array formula, at the very least.
And the -- forces the array of TRUE/FALSE into 1/0.

Both solutions are great ideas and work perfectly. Thank you very much.