Go for a huge traditional breakfast - with the haggis included, try to find it cooked in lard or tallow (some traditional places will still cook this way) and that will set you up for the day. Many hotels will have a full Scottish breakfast as a matter of course.
When I was in Scotland recently I survived pretty much on the breakfast and then ate fish in the evening. Scottish salmon fresh from the sea is amazing and you will find it everywhere, not just on the coast. Smoked salmon is gorgeous as a snack food and very popular as it is one of the traditional foods of Scotland (along with haggis). Available in supermarkets and local shops.
Aberdeen Angus beef is very popular in 'steakhouses' and restaurants, which you will find on most main streets in Edinburgh. Have a large cut with salad. Many places serve venison (deer) which is hunted on the Scottish Glens and sold in better restaurants, very worth trying.
Failing this, go for haggis and neep (cabbage) as a main meal. The haggis contains some oatmeal, but if that is the only grain you eat, then it is not the worst one you could choose. Many generations of Scottish Highlanders and Islanders were brought up fit and strong on a diet of meat, fish and oatmeal.
You will have absolutely no problem eating paleo when you are there. Although the typical Scottish diet is a SAD diet, the 'traditional' food is meat based, of which the Scottish are very proud, and if you are eating at more expensive places, the quality will be very good indeed.