Engineering costs for any project performed under this chapter, including the salaries or fees of the engineer, may be paid from the Local Bridge Replacement Rehabilitation Fund allocations to a county; however, the maximum fee paid to an engineer from the fund shall not exceed twelve percent (12%) of the final construction cost, with the methods of payment to be approved by the State Aid Engineer under regulations promulgated by the State Aid Engineer. No such costs shall be paid to an engineer before the letting of the project.