Who are the market makers for large traders?

Realistically to be large enough and savvy enough to be a "threat" to a market maker the trader will have to be an institutional investor such as a hedge fund, an insurance company or a mutual fund. They are the only market participants with the capital and analysis (from having large teams of analysts) to make a mark on an Investment Bank's (IB's) thinking.

Please note that the distinction between trade and position is very important below. The firm's position is the overall settled quantity that the firm has on its books. Trades are the result of an order being matched which change the quantity of the underlying position in the instrument. A firm will enter multiple orders that will be partially filled with trades to reach a desired final position.

Most large institutional investors have their own traders with Direct Market Access (DMA) and when they are trading large quantities they become the liquidity provider (note that I didn't say market maker as they aren't obligated to provide liquidity) for that instrument. In many cases they will take the other side of retail investors' trades and market makers who are trying to move their positions in the opposite direction. Another benefit to DMA is that they can strategically place a larger number of smaller orders to fulfill the price and volume requirements of the fund(s) that they are trading for. By "strategically place" I mean that the trader will place the smaller orders at a time and with a quantity where they can see liquidity already exists in the market. The funds that they are trading for want them to take a final position within a price range at the end of trading and don't much care what the traders do to get to that position. The success or failure of the trader is sometimes internally measured by the VWAP (Volume Weighted Average Price) of the final position. In these cases they are their own market makers. Trading large volumes will move the price of the instrument so the institutions' investors will enter orders in such a way, perhaps even over multiple days, that they minimize their impact on the market.

Part of minimizing their impact on market prices will involve setting limit orders that they expect to be filled rather than entering market orders. These limit orders will be close to "touch" (market price) if they need to take a position or at a price set by the analyst teams. orders on the other side of the book will partially match the volume of the limit order so the volume will be filled slowly over time instead of in one go. Where there is little liquidity for the instrument or the order volume is very high this will result in the price becoming a floor for the market price since there will continue to be volume at that price. Market makers will spot and exploit these floors either directly in lit markets where they can see the limit orders or by probing in dark pools. A large limit order sitting on the book is quite often a nice way for a market maker to offload a risk position. It is actually very rare that large volumes are traded through market orders as that is by far the most expensive way to trade.

Larger firms without DMA have the same incentives to drip volume into the market so won't normally need large volumes matched by market makers or the market. They also tend to have relationships with a number of brokers and market makers given their market presence. These result in the firms paying large overall fees to the market makers for other services such as market insights. The market makers' traders will take the longer view and help to fill the firm's position at a good price given that they will make up the loss on other fees.

If the trading firm is large enough to make a difference and unsavvy enough to place a market order the market makers will widen the spread far enough (and thereby move the market price) that they will make a profit from the trade and result in the firm having much higher trading costs. If the price changes sufficiently there will always be other participants who can profit as well. Market makers don't have to provide unlimited liquidity at a fixed price!