twinj, I think you may be confused.
A market maker is someone who operates mostly via limit orders. He therefore injects liquidy.
When you say you want to buy at the bid and sell at the ask to be a market maker, this has to be via (passive) limit orders. Then when a trader comes to sell his shares, he sees your bid quote and he sells his shares to you. Then in effect you buy at the bid, and the active/aggressive trader sells at the bid.
When you asked LMAX about buying at the bid and selling at the offer, I think LMAX thouht, you meant active (aggressive or any other name) buying and selling orders, in which you come to the market, see the orderbook and make your trade. You take away liquidity. In this case, you indeed buy at the offer/ask of the market maker and sell for the bid of the market maker.
So actually, you can be a market maker at LMAX. They offer you an API, which is a way to submit orders autmatically and quickly instead of clicking manually.
But, you have to note that Goldman, JPMorgan, and Optiver, are also on the LMAX. These firms both offer the liquidy by posting limit orders and they take it away via aggressive orders. The difference is that LMAX does not allow these firms to trade with each other. Thus they trade with retail traders only. They also have a much faster access to the computer of LMAX where orders are matched (lower latency than you, for sure). Thus, the most profitable arbitrage setups are pretty much out of your reach. You will be too late.
Additionally, you pay 0.0025% commission of the notional you trade. Goldman, Optiver, and JPMorgan most likely have a better cost structure.
In short, you can be market maker in some sense, though you will not have the priveleges as a real LMAX market maker.