It's not currently possible to set a specific weight to each product variation (modifier option) although it has been requested and in consideration as to the best way to present this as an option.
At present your two options are:
1) To create separate products for each weight variation. This way you can set a handling charge to each one if needed or just set the weight.
2) Create a custom extension using the extension hooks to set the shipping price based on user selection. This could be by way of an adjuster, a handling charge or just additional shipping charges.
The viability of the first one depends on the quantity of products and product variations you have. If it's a small inventory then this may be fine but for larger inventories it may not be ideal.