I am afraid that this level of experience isn't, and never has been sufficient for most airlines direct recruitment needs. The exception is those airlines that run cadet schemes usually in conjunction with one of the "big 3" training providers. Beyond those, there are a few companies that may give consideration to low houred candidates. Ryanair is perhaps the most notable, and somebody on these forums suggested that Jet2 may also fall into that category.
Beyond the recognised cadet schemes it is usually a case of "experience" and often that experience needs to comprise more than one part of the possible spectrum.
Instruction (if you can secure a job) is a useful way of building up hours quite rapidly, and whilst it is only one part of the spectrum, it has the dormant benefit of being useful within an airline career (training) many years later.
If you group "public transport" into a number of levels, at the top you have the prime (most coveted) employment opportunities. Below that you have your second and third tier airlines. Many succesful career climbers will have made their way up through these tiers over a number of years.
There are a very limited number of "fast track" routes into first or second or even third tier employment, but they are nearly always tied cadet programmes.
Far too many people have either convinced themselves, or been allowed to be convinced that a CPL/IR and a couple of hundred hours places them on the beauty pageant stage for airlines seeking First officers. It just isn't the case for the overwhelming majority of aspirants.
It is hard to see how a flying instructors job would in any way disadvantage you. It wouldn't prevent you from applying for anything else that might arise.
It would also provide experience, currency, and some income based on the profession you have broadly chosen. The alternative is sitting around as that "fresh training" rots like an apple in a bowl. Instrument ratings need to be renewed on an annual basis, and applications need a currency to keep them fresh.
It might well "take years" to get where you want to be via this route, but there may not be enough years in a lifetime to simply sit around in the vain hope that "straight out of training" wont become very stale very very quickly.
Airline recruiters are not "put off by extensive SEP experience" as many of them and the pilots they employ will already posess it. However going back to my "spectrum" comment, that isn't all that they are usually looking for. They want to see a progression through multi-engine experience and often a reasonable chunk of turbine time as well, depending on the current requirements at the particular tier you are applying to.
The best advice I can give you, is that experience (at any level) can only be a good thing. You then have the opportunity to move up to the next level. It may take more time than you originally envisaged, but sitting around waiting for the top slot, might be a very long wait indeed.