The whole point of the All Weather Portfolio (AWP) is that it provides both protection and growth opportunities in "all" (or at least many) economic conditions. I don't particularly think the mix in the original portfolio is the best, but it's an interesting starting point for a low-risk, long-term portfolio resistant to economic shocks.
If you are interested in this portfolio, you probably accept that economic conditions can't be reliably predicted, and want to sacrifice a little return for price stability and lower risk.
However, this only happens if you apply all parts of the portfolio. If you don't, then you essentially have the "Some Weather Portfolio", which will only give you protection and growth opportunities in some situations.
Stocks and bonds can continue to go up even after being called "expensive" by experts. The reverse can happen too. If you are able to reliably determine which asset classes are going to do better than others, then the AWP is not for you. Just go and buy the asset classes you determine to be cheap and sell the expensive ones. Rinse and repeat on each rebalance.
Of course, doing this is exactly what every active investor tries to do, and part of why doing this successfully is so hard. Most people, after all, fail at beating the market using this approach.
So if you accept the premises of the AWP, go all the way. If you want to try to time the market, do that instead (although I'd advise against it). Mixing the two might give you the impression that you are lowering your risk more than you really are, and hide the fact that you're taking on the role to forecast investment returns for each part of your target portfolio, even the parts you haven't invested in yet.