The problem is the Intel heatsink -- it is molded to hold the Intel fan and its plastic clips.
You might try getting a tip-magnetic fan the same size, or large enough that you could hopefully secure it to the aluminum heat-ink with sheet-metal screws to make an "interference fit."
Personally, I'd set the stock heatsink and fan aside completely, buy a ThermalRight SI-120, a Panaflo "High-Speed" 120x38mm fan in the "BX" series with TACH monitoring. You could probably run the Panaflo at 55 or 60% of its top-end of 2,500rpm -- and run it that way most of the time. If you have an ASUS motherboard, you could thermally control the fan through "Q-Fan" or in general you should be able to control the fan through your OS and "SpeedFan 4.2(x)" 4.26 has been released. It's free.
When I get heatsink-fans like the Intel stock processor, I save the sinks for a project I want to do later, and I use the fans and clip-on frame to make hard-drive coolers.
How much do you want for your stock Intel HSF? Postage plus $1?