I'm not sure that being a cash buyer has any advantages except they don't have to wait for you to get a loan approved.
If you finance it through someone (say a bank), they still get their money, the same as cash. If you don't finance it through them, then they may loose a loan commission if they're originating the loan for a lending institution. If they normally do the financing themselves, then they loose the loan interest.
You might ask if they'll lower the price if you finance it through them. If they will, and there's no penalties for paying off the loan early, finance it and pay it off next week.
Bill