So, you need to have a discussion with the head of engineering (not email; it needs to be real-time discussion, preferably in person, or at least over the phone) about this request. You need to explain that you are not going to provide the salary sheet because they should not have requested it in the first place **. You should ask him how that information is relevant to this process. And, if this person keeps pushing for it, then if you can do so in a non-combative way, ask them how, if the salary they are offering can change based on a factor that is not relevant to the job itself, then could the salary amount being offered also change based on your gender, age, race, etc? Those are also factors that are irrelevant to your ability to perform this job, and if they are going to be subjective on one level, can you trust that they aren't being subjective on those other levels? Skill and experience do affect how productive you might be, and so it makes sense to pay more for a more skilled and/or experienced candidate, but salary history does not directly correlate to either skill or experience, and so it should not be used to gauge such things.
So again, have a conversation to explain that you won't be providing it because you should be compensated for what you are going to do for them, not for what you did for previous employers. And if this head of engineering says that it is a hard requirement that you provide this proof, then you can politely cancel the interview process, explaining that such a requirement indicates to you that this is not a company that strives to treat its employees fairly, and that if this is how the interview process works, then you aren't convinced that, when it comes time for your annual review, that you will be evaluated based on what you actually did rather than some other arbitrary factor(s).
** Since you have already answered their question about prior salary, you might need to fess up about providing "inaccurate" information due to knowing how it would be used and not wanting being underpaid previously to cause you to be underpaid again. You can explain that part of why you are seeking a new position is due to being underpaid, and so you provided them with the amount that you should have been paid. And, if they are willing to pay X if you can prove that you are making Y, then they should be willing to pay X period, End.Of.Discussion.! Sure, you might be dealing with folks who have a narrow/immature world view who might themselves feel "cheated" to give someone a huge pay raise (in which case, do you really want to work for these people?). Or, conversely, you could be dealing with folks who have good heads on their shoulders and realize that paying people what they are worth helps when it comes to employee retention. Not to mention that in cases where paying someone what the job is worth results in a substantial raise due to the person being previously underpaid, that the person is highly likely to come in with a very positive attitude, get lots of work done, and appreciate the company enough to stay for a while.