One alternative would be to use a Bayesian method, as opposed to LASSO, factor analysis or principal components analysis. Bayesian methods do not have a problem with multicollinearity, unless, of course, its perfect collinearity. LASSO can be thought of as a Bayesian method with a Laplace prior distribution and a subsequent loss minimization procedure.
I find two difficulties with LASSO that you may not. The first is that I don't believe the implicit priors. The second is that I can only consider one possible likelihood function to generate the data. A Laplace prior around $\beta=0$ is a bit fanciful. Do you believe that these items do not interact with each other? While LASSO has an effective variable selection process, you cannot test other potential likelihood functions. Bayes would permit you to test both variables and probability models.
The final argument for Bayesian methods is that they cannot double count information. The reason for the absence of linear dependence assumption is that Frequentist methods double count information except when factors are orthogonal. In most cases, the impact is very small, but the fact that people look at variance inflation factors implies that it isn't always small enough.
When items strongly covary, they share an enormous amount of mutual information while possessing very little unique information. Once information has been included in a Bayesian calculation that same information gathered elsewhere will not impact the posterior as it is already in it.
There are two ways to calculate a Bayesian posterior. One way is to consider all the sample information jointly in one grand calculation. The other is to consider the sample one observation at a time. The prior would normalize the likelihood of a single observation and produce a posterior. That posterior would become the new prior distribution and the same process would happen for the second observation and so forth.
Visually, if you watched the shapes of the posterior evolve as individual items from the sample were included, you would begin with a very unstable, broad and sometimes quickly shifting geometric entity. Assuming a very large sample, you would notice the posterior barely altered with each new observation as Bayes theorem is ignoring any information it has already captured.
I produced the posteriors for a sequence of coin tosses. I was going to do a bivariate, multicollinear regression problem, but then I realized that it would take hours for the Bayesian side of the simulation to run and I don't have the free time. If I can get a bit of free time, I will edit this post with a visual simulation of OLS versus Bayesian regression sampling distributions. It won't be this upcoming week though.
To show the decreasing size of the impact of information from observations, I simulated two hundred coin tosses. The parameter was believed to be larger rather than smaller, so I used a Beta(2,1) prior density, the triangular distribution. It helps to see the impact of prior knowledge on the parameters estimates.
Graph one is that of experiments one to five with the prior included so you readily see how the posterior wiggles around with each additional piece of information. During Rounds One to Five, the Prior has a substantial effect because it is equivalent to observing two successes and one failure, before the first coin toss.
Graph Two is for experiments five through two hundred. As you can see, between rounds twenty and two hundred, the mode is no longer moving around much and while the mass is getting narrower, it is nothing compared to the narrowing from early in the sample, which you can see better in the plot of all posteriors.
The complete set hopefully makes it a bit more clear how it takes many observations later in the sample to make a small movement when compared with early in the sample.
When there is no mutual information, this same convergence happens with Frequentist methods as well, but there is no comparable construction for Frequentist methods except possibly to show the movement of the boundaries of a confidence interval during a series of meta-analyses.
I will try and build a simulation to show the effect of the double counting graphically. Going back to the earlier discussion, priors matter and LASSO has a very pessimistic and unrealistic prior, that there is no effect. Of course, that is the reason that Bayesian methods are considered "optimistic" methods and Frequentist methods are considered "pessimistic" methods.