First, if he is applying for permanent residence, he wants to avoid staying in Canada without authorization (without a valid work or visitors permit). Overstaying is grounds for refusal, so you want to avoid letting the permit expire.
HOWEVER - after a permit expires, there is also 90-day "grace period," during which he can apply to "restore" status (extra fee, $$$) along with applying for a new work permit (or visitors permit). This may be an option if his employer does apply for an LMIA, but has to wait during processing. The downside is that (a) he cannot work after his permit expires, and has to wait until he gets a new permit to resume working. Also, if he waits for the LMIA and goes out of status, and the LMIA is then refused, his options are very limited.
Third - Some people have tried this and succeeded, but I'm not necessarily recommending this, it is a gamble - After applying for the CEC program, he can submit the BOWP application on paper, which takes longer to process than submitting the application electronically. As long as the application is submitted BEFORE the old permit expires, he will be on "implied status," meaning he is authorized to continue working until he receives a response to the work permit application. Because the paper version takes longer to process, sometimes the AOR for the CEC application is ready by the time that CIC starts to process the BOWP application. Sometimes this works, sometimes it doesn't - and it seems that AOR processing times are getting a bit longer, so it may be less likely to work in the future. And if he gets a refusal, then he is out of status (and the 90 day grace period starts...)