Publishing a board game with an existing game company
That deal doesn't seem very likely. You plan to cover any expenses they may incur during production, marketing, and distribution, but I do not believe you have considered their opportunity cost for taking on the project. Whatever time and resources they are devoting to your game (even if you are footing the bill) could be spent elsewhere on a product with a greater expected return on investment.

It sounds like you are looking for the board game equivalent of vanity publishing (https://en.m.wikipedia.org/wiki/Vanity_press), which is usually not a good idea for anyone except the publisher. If you do find a company willing to let you cover all costs, it may not be as good of a deal as you think.