There is no point you contacting PIP. The delay is down to the assessment provider. Step 1 is follow the provider's complaints procedure. Complaints get a very (well, relatively) quick response. The contact info and guidance will be on their website, if you know the name of the company (atos or capita depending where you live), I can link it.
Complaints structure is as follows: what happened, what went wrong, what they can do to fix it.
What happened : you are supposed to be reassessed by them for your pip
What went wrong: unfortunately you have had seizures during multiple assessments meaning they have had to be concluded, this is not your fault. Not only this but your mum was told she could not answer questions about how you are following a seizure even though you couldn't possibly answer this yourself, due to your health condition, and wanted her to help you answer. This is contrary to their own guidance on companions at assessments. As a result you face a long delay in having another appointment due to their failure to rebook you a suitable (home?) appointment, meaning your fixed term PIP award will end before a decision is made about your new one, and you are stressed about being told you have to answer questions when you will struggle to do so and brought someone with you for that reason. The delay in rebooking is unacceptable and will leave you in financial hardship as a result.
What they can do to fix it: urgently review the evidence you have already provided along with the partial assessments conducted and decide whether they can now make a recommendation on your entitlement without seeing you face to face. Alternatively, fast-track another assessment, preferably at home where you will be less stressed and therefore less likely to suffer a seizure, to ensure you are not waiting an unreasonable amount of time.
Step 2 (simultaneously) take it to your MP. In this case the assessment provider are being unreasonable not rebooking you as soon as they can, however they may genuinely not have any quick appointments and that is where the MP may come in handy in escalating the matter to someone higher up who can arrange that. Your MP may however say their hands are tied because the assessment provider is not part of the DWP but a private company. This is true. However MPs are perfectly able to advocate for constituents dealing with private companies and no reason why not.
NB if your new pip award is not decided before the old one runs out, due to all of this delay, I think it should be backdated so there is no gap but am not 100% sure if that is how it works for renewals.
For general info: to get the assessment provider to see you at home, realistically you need a doctor's letter, ideally sent off with the health questionnaire / PIP form rather than later on down the line, explaining it is not safe for you to travel to assessments for whatever reason, and also to write this separately on the form itself.
Finally, the process of "getting someone to speak for you" sounds like they are actually trying to suggesting your mum should be your appointee. This is entirely different from the assessor being willing to listen to her at a medical assessment. An appointee receives your money, signs your forms, and gets your benefit letters. It is for people who, for whatever reason, can't manage their own claims. This may well suit you, but just for your info.
It is true that the assessors' guidance states that the assessment should be primarily conducted with you as the claimant, rather than a companion, but it does not mean she has to be your appointee to help you at the assessment. I've pasted the info below:
Consultations should predominantly be between the HP and the claimant. However, the companions may play an active role in helping claimants answer questions where the claimant or HP wishes them to do so. HPs should allow a companion to contribute and should record any evidence they provide. This may be particularly important where the claimant has a mental, cognitive or intellectual impairment. In such cases the claimant may not be able to give an accurate account of their health condition or impairment, through a lack of insight or unrealistic expectations of their own ability. In such cases it will be essential to get an accurate account from the companion.
1.6.54 However, the involvement of companions should be handled appropriately by the HP. It is essential that the HP’s advice considers the details given by the claimant and the companion and whether one or both are understating or overstating the needs. If the presence of a companion becomes disruptive to the consultation, the HP may ask them to leave. However, this should be avoided wherever possible.
1.6.55 HPs should use their judgement about the presence of companions during any examination. A companion should be in the room for an examination only if both the claimant and the HP agree. Companions should take no part in examinations.
1.6.56 The presence and involvement of any companion at a consultation should be recorded in the assessment report.