This seems possible in principle, by replacing the oligo(dT) in the
1S primer with a sequence complementary to your target, though we've
tried it before with poor results.
The annealing kinetics of the first-strand RT primer is probably
going to be the least of your problems, because of both the
temperature and the unusually high ionic strength in the TS-RT
reaction - any primer long enough to have a specific sequence is
probably also long enough to anneal. However, that specificity is
the bigger problem. The normal version of the protocol uses
oligo(dT) to select for polyadenylated RNAs, i.e. to avoid priming
from ribosomal RNA, which is the vast majority of total RNA. So
you'd want to be very sure your custom primer has no similarity to
your species' ribosomes. Most likely the vast majority of the RNA in
your sample is not your transcript of interest, so there's also the
challenge of avoiding off-target priming on other transcripts,
especially those that are more common. And if the abundance of the
target transcript is low, you might even be competing with potential
priming from genomic DNA via strand invasion. So you should probably
treat your sample with DNase, and possibly even an initial rRNA
depletion if that signal dominates your data, but regardless you'll
probably need to use a lot more RNA because you're aiming for a much
smaller target.
A way to go about it might be to test your target-specific 1S primer
with large amounts of non-precious input RNA known to express your
target transcript, and compare that with negative controls: a
no-template control, just to check for primer dimers (which might be
substantial), and if possible also a similar RNA sample that does
not express your target transcript (e.g. a knockdown), to check for
off-target priming from other transcripts. You'll definitely have to
reoptimize the relationship between input amount and PCR cycles, and
you may also have to reoptimize the concentration of the 1S primer.
You can simplify the latter optimization by adding SYBR Green or
similar to the library PCR and running it to completion in a qPCR
machine; you'll need to watch not just the absolute yield but also
the relative yield above the negative controls. When we tried this
before, we saw high yields even from the no-template control,
presumably primer dimers, so even with 500 ng of total RNA we
weren't actually getting more library than we already did with no
RNA. Of course this will depend on the target sequence. You might
also be able to reduce the formation of TS-RT primer dimers by
removing the random bases from the 2S primer, since you'll need to
spike in a lot of PhiX to get base diversity for Illumina sequencers
in the rest of the read anyway.
Aside from those problems, you might have some flexibility in other
parts of the design. Instead of the 3' end you could aim your primer
at some comfortable distance from the 5' end of the transcript (e.g.
400 nt). Then your amplicons would already be the right size and you
could skip the initial fragmentation. But it might still help to
denature the RNA in case there are secondary structures that inhibit
RT, so you could keep that first incubation and just move all the
magnesium ingredients (which cause fragmentation) into the later
step as we do in the FFPE LCM protocol. This could help you exclude
off-target products by a more stringent size selection, and
depending on your goals you might also want to keep the cDNAs long
anyway so you can read more of the sequence.
If that doesn't work, another approach is to make the initial cDNA
from all polyadenylated RNAs as usual, and then add the target
specificity (and one of the adapters) during PCR instead. Then you
wouldn't have to worry so much about TS-RT primer dimers. That
appears to be the strategy used in the SMARTer TCR kits from Takara
Bio.