Structure it so that closing is triggered by visa issuance. Funds go into escrow before the interview, and the escrow releases on approval, so the buyer is neither uncommitted nor stranded with a business he cannot enter to run.
Let the escrow date, not the seller's impatience, set the closing
A buyer who closes first and applies later has placed funds at risk, which the E-2 rules require, but if the visa is refused he owns a U.S. business he cannot direct from inside the country. A buyer who applies first with nothing committed will be told the funds are not at risk.
The standard solution is a purchase agreement that closes on visa issuance, with the price in escrow beforehand under instructions that release it only on that event. Sequence the work accordingly: complete due diligence, fix the licence arrangement, sign the agreement with the escrow clause, fund the escrow with traced money, prepare the business plan, then request the consular appointment. Interview scheduling varies by post, so ask about current wait times before the seller sets a deadline.
If the seller will not accept a conditional close, the buyer must weigh the risk of owning an unworkable business against the cost of walking away.