Hiya,
I agree with Smack - internships are usually intended for undergrads. That said, I think it'll be good to not place your eggs in one basket and A/B test applications for both grad schemes and internships, to understand which yields more response.
Of course, invest the majority of your efforts in applying for grad schemes, but it'll be good to apply for internships too, as grad schemes are usually incredibly competitive.
I think another thing to note also, is that you may enjoy an internship at a certain firm, more than a grad scheme at another, and the internship may potentially open up more doors for you, due to what you learn/the company name etc, than the grad scheme. So don't dismiss an internship, if it's something you enjoy, due to a shortfall in perks, but weigh the pros and cons.
I think that you have organised your job hunt very well, by creating a short list. I would encourage you to continue to show your passion, and jump at any opportunity that will get your foot through the door. Employers understand not to expect a lot of work experience, at a graduate level, and are often more focused on your skill set and what else you can bring. You can build skills that are required for jobs, through several other activities and commitments. It may therefore be a good idea to build on your skills, be able to demonstrate these through experiences/achievements and make yourself more employable.
If you have any specific questions, or would like more career advice, please feel free to DM us, and we will be more than happy to discuss
Milkround Graduate Careers Team