Edit: Most of this answer is based on a part of the question that has now been removed:
Assuming the above is illegal, then I think a Limited Partnership, or a Limited Liability Partnership may be the answer[...]
I actually work in an almost identical situation.
You don't have to register a Limited Partnership or a Limited Liability Partnership (LLP)—you can register as an ordinary business Partnership—which is very similar to being a sole trader and you are still technically self-employed. Similarly to being a sole trader, you are still personally responsible for all losses, bills, debts etc. But all partners are responsible for their share of the business.
When you register as a partnership, one partner will be the nominated partner—who is responsible for submitting the partnership tax returns and keeping business records.
You can both take your share of profits, pay your own NI contributions and any tax due on your share of the profits etc, essentially exactly the same as being a sole trader. The only difference is that you have to submit a separate tax return for the partnership.
You can even do—as you said—have all the income come in to your bank account, pay your partners share to him and account for it as business income and your respective shares of the profit, without the need for invoicing each other etc.
Relevant Documents & Reading (GOV.UK):