The short answer is no.
Here's a review article with some evidence:
B cell receptor editing in tolerance and autoimmunity
Section: Receptor Editing in Mice with Conventional Autoantibody Transgenes
If multiple κ rearrangements fail to yield a suitable L chain, the κ
locus undergoes rearrangement to a non-coding gene called RS in mice
(RS is referred to as the κ deleting element in humans). RS
rearrangement results in either deletion or inversion of Cκ,
consequently inactivating κ L chain expression on that allele. In cell
lines with λ rearrangements, one or usually both κ alleles are
deleted, whereas in κ-expressing B cells the fraction of RS
rearrangements is approximately 15%.
In summary: After a point with (probably multiple) κ rearrangement failures, both κ loci (usually) get deleted, leaving only λ. So if a B cell expressing a BCR with λ is self-reactive, it won't have the chance to rearrange its κ.
However, there's a lot more involved. For one, it's possible to get a functional and non-self-reactive λ chain BCR without repeated κ rearrangement failures.
Since the quoted passage might be confusing out of context, I'd like to mention briefly that your question's description of light chain recombination is very simplified. In reality, there are many additional mechanisms involved and many different paths to get a functional κ rearrangement, a functional λ rearrangement, neither, or even both. Moreover, there's a lot we don't know, since we learn one experiment at a time, and we can't always directly measure what we'd like to.
First, rearrangement has been observed to occur simultaneously in κ and λ. This fact might explain why it's possible to get a functional λ chain BCR right away as mentioned above. Second, allelic exclusion (between both κs and both λs) doesn't always work as you and I might expect it to, and a developing B cell can express different BCRs simultaneously (even one with κ and the other with λ). Third, due to receptor editing, a B cell has multiple opportunities for a functional κ chain rearrangement on a single chromosome, meaning that neither λ nor the second κ is needed despite an initially nonfunctional κ rearrangement.
I highly recommend reading the review article!