With three places at the Open Championship up for grabs, along with a first prize of £170,000, the leaderboard is bunched at the European Tour event, with nine further players within two shots of the leaders.
For Lee, there could be no greater incentive, having never played in a major championship, and he followed his opening 65 with a four-under-par 67 on the Royal Johannesburg & Kensington club's West Course.
That took the 36-year-old long-time qualifying school regular to 11-under 132, having played his first round on the par-72 East Course in seven under par.
"The goal was to not fall back and there's a strong field behind you, so it was important that I didn't slip back," Lee said.
Molinari began his week with a seven-under 64 on the West Course, and followed it up with a four-under 68 on the East Course to stay out in front.
In closest attendance to the leaders after 36 holes were England's David Horsey and South African George Coetzee, one shot back.
Horsey enjoyed an eight-under 63 on the West Course and is relishing the battle for Open places, as well as this weekend's title.
"It's an added bonus," Horsey said on europeantour.com.
"I think it's a good system, a good change from the R&A to do that. It makes the qualifying a bit fairer and spreads it out over a longer period rather than making it a 36-hole sprint. It's at the back of my mind at the moment, but it's certainly a bonus to consider near the end of the week."
Finland's Roope Kakko posted an eight-under 64 on the East Course to lie on nine-under 134, in a seven-player share of fifth place that also included England's Danny Willett and James Heath, and Scotland's Alastair Forsyth. Forsyth added a steady 70 to his impressive opening 64.
Yorkshire's Simon Dyson is among those who cannot earn a place in the Open at Royal Liverpool through this tournament, having failed to make the cut.
Dyson posted back-to-back rounds of 70 to be three under, but he needed to pick up another shot to make it through to the weekend.