Instead of having it so that lightweight always gives you 10% damage taken increase, make it so that the extra 10% damage taken debuff is only active when you are holding the weapon.
For example, if you have a primary without lightweight and a secondary with lightweight, make it so that if you switch to your primary you are back to normal speed and normal damage taken. If you switch to the secondary, make it so you have the speed boost and the extra 10% damage taken debuff.
For example, if you have a primary without lightweight and a secondary with lightweight, make it so that if you switch to your primary you are back to normal speed and normal damage taken. If you switch to the secondary, make it so you have the speed boost and the extra 10% damage taken debuff.