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Your question as is doesn't give a lot to work with.

Nobody knows what company or country you're in, and "software engineer Mk II" doesn't mean anything to anyone. I don't even know if you applied for the raise or they just gave it to you! Did you agree to the promotion they offered, and was there dialogue then?

Natually, you won't want to disclose your company, but you might want to consider expressing the raise as a percent of your current salary - something to get a feel for what you're asking for. Also something like "i have x years work experience" will help too.

Nobody knows what the responsibility change means either - are you working longer hours? Travelling more? Or same hours but now if something goes wrong you look bad?

I'll tell you, you're going to struggle with an email, or indeed, any sort of bargaining. You're much better off taking this raise to senior engineer and then shopping around at other companies to get the raise.

Then you can either hop to the other company, or ask your company for a raise based on the new job offer that you currently have.

Absolutely do not reject a raise because it isn't enough money - they're giving you more money, take it. Use that to earn more money elsewhere. Rejecting the raise (on the grounds of money) is going to raise red flags in your company, and not help you find a better paying job.

You're actually in a better spot than most people on the promotion track - typically it's the reverse. You get the increased responsibility and then, after a year, you get the raise. Maybe your company is going to up your salary after you've proved you can do the job? Whatever it is, I'd strongly advise against declining more money.

Well, almost. If you're being asked to do work that is hindering with your lifestyle (eg longer hours, travel, stress), and you're not getting compensated for it, and you never wanted it anyway, then it is fine to say "no thanks".

If you wanted it and you're just not happy with the amount (and where did you get that 20% figure from anyway?) then just take it and look for other jobs that pay more.