Investors are limited to purchasing from the trading volume of bitcoin (or possibly a higher amount?), you can't determine the maximum amount of bitcoin they can buy from its market cap because a large percentage of bitcoins are not for sale and locked away in cold storage.
The trading volume of bitcoin according to Coingecko is $13 billion, that must be why institutional investors are limited to backing their funds with hundreds of millions worth of bitcoin. Which leads me to think that it isn't feasible for a fund to pump the bitcoin price by itself because it would never be able to secure $13 billion of bitcoin from a single exchange. They'd have to buy up lots of bitcoin from several dozen exchanges and regulatory channels exchanges place will inhibit funds from buying more than a few hundred thousand $'s worth of of BTC.
I would assume that if they ever want to buy 13 billion dollars worth of bitcoin, they could do it OTC and they wouldn't really have trouble with it. Sure they may not get it at one place, but for example if a huge company wants to find 13 billion dollars worth of bitcoin somewhere, they could track down who has the most bitcoins and mainly that is the exchanges themselves and not the users of those exchanges.
So, if you want 13 billion dollars worth of bitcoin, you contact bitfinex, bittrex, poloniex, kucoin and mainly binance when you combine all the bitcoins they have you can definitely get a lot of bitcoin from them, include couple of big whales in there as well and you got yourself 13 billion dollars for sure. Obviously on a market where the volume is 13 billion, you will increase the price a bit but at least you can do it all on OTC.