Additional context: We're using Bootstrap v2.x and compile our css using the less files.
For an existing app, you can't just remove bootstrap and put in something new. As you probably realize by now, everything will just break.
Instead, you need to update individual pages along the way. This process is preferable to removing bootstrap and re-implementing your pages all in one giant pull request since it allows your team to QA iteratively.
For us, we redesigned our app, made it responsive, and also used another library in addition to removing Bootstrap. There was a lot more nuance involved, but for example, I started first by redoing the header and footer, then I moved onto individual pages. You can probably just start by re-implementing common mixins and modules first. Then you can move onto the actual templates and start removing your .rows and .span divs page-by-page.
I wrote about the process in depth here: https://medium.com/anyperk-product-engineering/anyperks-resp...
However, your login button appears to be wrapping
Edit: Chrome 42 - OSX 10.10
Another source of annoyance might be a large number of unused rules. I believe you can just remove the individual LESS files that Bootstrap provides that you don't want. I personally use about 4 of the 10+ LESS files that Bootstrap offers (at least for a vanilla project).
Approach it like you would any rewrite, tackle each part of your application as a separate modular component. Do the homepage, then static pages and so on until you have moved every page over from Bootstrap.
Why haven't you upgraded to bootstrap 3? It could be a bit lighter when you consider the mixins for grids that it provides.