Copyright is often referred to as a "bundle" of rights. The UT Libraries' copyright page has basic information about the four rights that make up copyright.
A copyright owner can give anyone particular permissions to their work. A publisher needs only the copyright owner's permission to reproduce and distribute the work in order to publish it. The publisher does not need to be the copyright holder in order to publish your work. Consider this statement from Kevin L. Smith, J.D., and David R. Hanson, J.D., in Copyright and Author's Rights: A Briefing Paper:
"Some publishers use a 'license to publish' agreement instead of a full transfer of rights... Even when publishers initially demand a transfer of rights, many have already-prepared 'license to publish' agreements available if the author requests it."
There are many examples of journals who do not ask for copyright transfer as a condition of publishing with them (e.g., Evolutionary Ecology Research or PLOS Medicine).
As a result of many funders' public access policies, publishers are regularly accommodating requests to amend or alter traditional copyright transfer agreements. And, with several universities passing public access or open access policies at their institutions (Harvard, University of California system), publishers are increasingly familiar with fielding author requests to amend default publication agreements.
So, you aren't the first person to ask for a change in publication agreement terms.
First, consider what you want to be able to do with your work without having to ask the publisher's permission.