What you propose is not a good strategy on several fronts. You cannot register a domain name similar to your competitor and expect to get rankings for the words in that domain name.
Redirect domains don't influence rankings
Redirecting a domain name to your site doesn't have any influence with Google. The only time that Google might increase site rankings for a redirected domain would be if that domain has lots of external links pointing to it. However, a freshly registered domain name isn't going to come with links at it would be a better use of your time to build links to your own site rather than to a redirect.
Exact match domains don't make better rankings
Even if you re-branded your site with this domain name it likely wouldn't help much. Google changed its algorithms a few years ago to demote exact match domains in the search results.
What you are proposing is trademark infringement
Besides the fact that it won't work, what you are proposing could get you in legal trouble. Even it it isn't registered, your competitors have a common use trademark on their name. Registering similar and confusing domains is exactly the type of thing that trademark law was written to prevent.
You say that you "don't want to mislead people", but you are proposing ranking your home page for your competitor's name. That is precisely the definition of misleading people if you ask me.
If you want to legitimately rank for a competiors name you could create a page that compares your services to theirs.