I do not use Final Cut Pro, but I play one on Television! 😁
Here is a quote from Wikipedia:
In April 2004, version 4.5 of Final Cut Pro was introduced and branded by Apple as "Final Cut Pro HD" due to its native support for Panasonic's tape-based DVCPRO HDformat for compressed 720p and 1080i HD over FireWire. (The software had been capable of uncompressed HD editing since version 3.0, but at the time had required expensive video cards and high speed storage.)
Any program written in 2004 would have been a PowerPC application. After Apple migrated all of its future Macs to the Intel platform in 2006, it included in OS X Tiger (and later Leopard, and optionally in Snow Leopard) emulation software that would allow most PowerPC apps to continue to run in Intel. This sofware was based upon technology that Apple licensed from a 3rd party and which Apple called Rosetta.
Apple's license to include Rosetta in newer versions of OS X expired with Lion and now Mt. Lion.
Hence no matter how well you use Terminal, or even Pacifist to install FCP 4.5, it just will NEVER run under Lion on your Mac.
If your MacBook Air 11" is old enough (2010, MacBookAir3,1) to boot and run Snow Leopard, you can either partition your hard drive or add an external hard drive and install Snow Leopard into it. You can then "dual-boot" into Snow Leopard when you want to run FCP 4.5 or into Lion when you need iCloud or otherwise.
Alternatively you can install Snow Leopard Server into Parallels 8 on any 11" MacBook Air and have FCP 4.5 running concurrently with Lion, such as here where the PowerPC app Appleworks is running on a Lion Mac:

[click on image to enlarge]
Apple is now selling Snow Leopard Server for $19.99 + sales tax & shipping costs at 1.800.MYAPPLE (1.800.692.7753) - Apple Part Number: MC588Z/A (telephone orders only).
Parallels offers a 14 day free trial download from its website. It sells retail for $79 but can be found for a discount on the internet.
NOTE: You are confusing your version with Final Cut Express 3.5 and Final Cut Studio 2, both of which will run in Lion, but whose installers have problems in Lion and Mt. Lion due to a "bug" in the installer program. For those versions, as pointed out in Jeremy Johnstone's blog, using Terminal (or alternatively, using Pacifist) to install will achieve success in using those versions in Lion.