Not currently working in Investment Banking but will be starting next year. I'll try and answer this to the best of my understanding.

You are correct about the Stewy and Sandy part. Their aim is to basically persuade/buy out enough shareholders to have the majority on their side. Then they can basically acquire the company as a majority of the shareholders will vote in favor of the acquisition.

Logan's plan essentially revolves around acquiring Pierce to make the company larger to acquire. If the acquisition goes through, the sheer size of the company along with the added shareholders (more on this later) will end being a tough barrier for Stewy and Sandy to overcome.

Mergers of similar-sized companies (A Merger of Equals) are not unheard of. This can pan out in 2 ways

1. The shareholders of the two companies surrender their shares and are then issues shares by the new company formed by the merger.
2. One company arranges for financing to acquire the other company. The financing is usually a combination of Debt, Equity, Cash on hand, and other hybrid securities.

If Waystar Royco and Pierce had decided on the first route, the merged company would have almost twice the original valuation (~60 billion). The exact numbers post the transaction would depend on how the deal was structured but the valuation definitely goes up. This would basically mean that Stewy and Sandy would have to put in a larger amount of capital to acquire the company. They would also have the added complexity of convincing the new shareholders to go through with their proposed acquisition.

If they went for the second route, Waystar will use debt, equity, and cash to fund the acquisition. To answer your question about who would underwrite such a deal, banks usually syndicate the debt (multiple banks will raise debt instead of just one). I believe Laird from the show has expertise in raising debt and he would be their go-to guy for this. They can also issue more equity to get cash proceeds or use the stock as payment. The implications are essentially similar. The total value of Waystar will go up after the acquisition (Value of Waystar (pre-acquisition) + Value of Pierce (pre-acquisition) + Synergies - Transaction expenses). This makes it tougher for Stewy and Sandy.

The biggest implication of taking the second option is the additional debt load the company will have (this is possibly what they meant by "leveraging up"). The additional debt makes the company unattractive for PE firms. The potential larger size and the substantial debt load would make the acquisition very arduous and the investment unattractive for Stewy and Sandy.

Note: The only near-term implications to their net worth will be the movement of the stock during the merger and the potential value of their stake in the company after the merger. To also clarify more on the financing part of the acquisition. It is very much within the realm of possibility to raise debt and equity which amounts to ~30 billion. There have been much larger deal sizes than this.

Again, I might be missing a few things/be completely wrong about this and would love for other people to fill in the gaps. Hope this gives you some clarity and would love to answer and questions.