Regarding two different aspects of the proposed bridge:
(1) build from 2x8's or larger dimension treated lumber what would look like a stud wall, covered on top with marine-grade 3/4-inch plywood topped with flat thin sheet metal.
Leave the bottom face of the entire assembly open for drainage.
Space the 2x8's close together, perhaps a foot apart.
Block and brace sideways between the 2x8s.
Watch the sheet metal as will be slick when wet.
We did something similar and washed the sheet metal with vinegar, let dry, dumped a gallon of cheap enamel on it, rolled it out, threw on sand, let dry. Repeat every five years or so.
(2) regarding the Corps of Engineers and their jurisdiction over what are called 'waters of the United States':
If you want to do it by the numbers or if in doubt call your local Corps District office (search on the web for Corps HQ) and ask for a 'No Permit Required' for "404(f) Agricultural Exemption". If asked, be prepared to explain what your "agricultural" activities are (hay, fruit or nut trees, silvaculture, aquaculture, livestock, grazing). "Agriculture" covers a multitude of activities and you can be somewhat imaginative: tie a goat to a stake on the other side of your creek.
When talking with your local Corps District you can ask for information without giving your name or address and see what they locally think about the applicability of the 404(f) exemption. You can also ask for a "PreApplication Consultation" by phone or in person and tell them as much or as little as you feel comfortable with.
Mention is made of a 3-ft setback from the creek bank. I'll bet that wasn't the Corps. The Corps jurisdiction is tightly defined by law (Clean Water Act Section 404) and 3-ft back of the bank is NOT Corps jurisdiction.
If you place NOTHING structual (piers, columns, bents, gravel, rip-rap, fill) down between the banks (staying back of the bank) or into the water (technically, the 'Ordinary High Water Mark') you do NOT need Corps approval as you would be outside their jurisdiction.
If you wanted to place a culvert (properly sized for hydraulic flow at that unique location) you would qualify for a Nationwide Permit #14 for Transportation which you could SELF VERIFY use of the Permit (a process written right on the first page of the Permit under 'Notification'---the Notification tells you when you must contact Corps; if you do not fit that criteria, you are NOT required to contact the Corps and you may Self Verify use of the NWP and NEVER contact the Corps.
In fact, you could SELF VERIFY the NWP-14 for your bridge and NEVER contact the Corps. Except you may qualify for an Agricultural Exemption and NOT need any Nationwide Permit.
My suggestion is to
(a) legitimately claim 404(f) Agricultural Exemption (also via Self Verification, see below how to document) and if you choose request a written 'No Permit Required' response from your local Corps District or
(b) Self Verify a NWP-14 and tell the local busy-bodies you're legal. They'll b**** and gripe but the Corps will support you.
If you choose to Self Verify, print a copy of the 2012 Nationwide Permit Number 14 for Transportation, write yourself a letter stating you believe you are acting in good faith after having read the Permit and understand its terms and conditions, sign it, mail it to yourself, save the postmarked envelope, staple it all into a package, stick it in a file, and forget about it until somebody bothers you. Then drag out the signed dated Permit and letter and tell them you are fully legally permitted because you are.
The Corps is generally pretty easy to work with for small landowners.
That said, you may have in your area other 'Authorities Having Jurisdiction' (AHJ) such as cities, municipalities, townships, counties, parishes, flood plains administrators, emergency management, road and bridge districts, road commissioners, quasi-governmental environmental groups, a multitude I couldn't begin to identify at a distance---each one of which may get into your business in some form or another.
Although in a different geographical area, I have a bit of practical experience in exactly what you're proposing to accomplish.
Post back if additional information is desired and how you choose to proceed.