Count the protons. Figure out the amount of pure methanoic acid. Consider the addition of the dilute acid to your mix as that amount of methanoic but include the water in the total volume. Consider it as pure water. Do the same with the lye i.e. find out how much pure lye and how much pure water are being added to the mix.
Guess an answer pH. Calculate the concentration of OH- and H+ at that pH and determine the number of protons that you would have to add (or subtract) to your volume of pure water to reach that pH. If your guessed pH is < 7 the H+ concentration will be higher and you will have to add protons to establish that level and neutralize the 10^-7 concentration at pH 7. Call this need for protons the "proton deficit". If the guess pH is > 7 the OH- level will go up and the H+ concentration goes down and the H+ concentration of 10^-7 at pH 7 represents a surfeit.
Assume the NaOH is completely dissociated. Thus you will need as many protons as there are OH- ions in the added NaOH to convert the OH- to water and the eqivalents of OH- become part of the proton deficit.
Now the tricky part. Use Henderson - Hallelbalch to compute the charge on the formic acid ion at your guess pH. To do this compute the dissociation ratio
r = 10^(pH - pK). This is the ratio of dissociated molecules to undissociated ones. If there was 1 mole of formic acid to start with and x remain undissociated the rx must be dissociated and x/(x + rx) = 1/(1 + r) is the fraction dissociated. The original acid is undissociated so in being added to a solution which has a certain pH it dissociates supplying C/(1 + r) protons where C is the number of moles of formic acid. This is a proton surfeit (with respect to the guessed pH).
So, assuming the guessed pH is < 7
1. Protons are required to neutralize the OH- in the pure water (deficit)
2. Protons are required to establish an H+ concentration in the water of 10^-pH (deficit)
3. Protons are required to neutralize the OH- ions from the NaOH (deficit)
4. Protons are released by the methanoic acid (surfeit)
Since no protons are generated or destroyed in this reaction the total of the deficits must equal the total of the surfeits when the reaction is complete. All you need to do is keep guessing pH's until you find the one that zeroes the sum. The easiest way to do this is to put your calculations into an Excel spreadsheet and use the Solver to find the pH. If you don't know about Solver, learn about it! If you don't want to it's pretty easy to zero in on the correct answer pretty quickly manually.