HI scootbabe - I have 30 years experience in fostering & adoption both as a social worker and manager for a LA and am now retired.
I think you ae trying to run before you can walk.....don't mean that critically really. When you approached the LA and the iFA were you thinking of short term fostering (thought that can sometimes stretch to 2 yrs and beyond) or permanent fostering? I think BMP confused you by saying they wouldhave to do the assessment. This isn't how it works. Your own LA would do the assessment and then they would "sell" you to the London Borough (it's called an inter agency agreement) and it would proceed on that basis. The child's sw in London would then decide whether you were a god "match" for this child. Mostly it is only approved adoptors that contact the journals, showing interest in some child featured.
I realy don't think it's a god idea to pursue any interest in the child you saw advertised in the BeMyParent website. The reason I say this is because you can tell very little just looking at a picture and reading the profile, which will not give you any real idea of what this child is like. Also the children who are advertised in these journals are all children who the LAs have been unable to place, and so they are widening the search nation wide, and so they are likely to be very difficult children. I know it's tempting to see a cute looking child but believe me there will be mega problems associated with the child and his/her past experience. This is also the case with LA children too.
My advice (FWIW) is to decide whether you want to pursue your application via the LA or the IFA. What age are you considering? I think the best thing is to pursue your application via the LA and the reason I say this is because LAs only use IFAs for children they cannot place in their own authority (for short term placements) because IFAs charge the LA a significant amount of money for carers that they have approved. If you decide to go with the IFA, then ask how many children of whatever age range you are thinking of that they have placed in the last 2 years. The thing is LAs are all cash strapped as this govt have slashed the budgets of all public services, so they really can't afford to "buy" approved carers from IFAs - so you will probably find the IFA ready to pursue your application quicker than the LA because they have much more time and more resources that LAs do but if they approve you, as I've said before you will have to wait for the LA to need to "buy" you from the IFA.
The other thing is that you are always invited on to a preparation group which covers all aspects of fostering and gives you the opportunity of learning far more about fostering and raising any queries that you might have. The assessment follows if you still wish to proceed. LAs run these courses through the year but you may have a wait for the next one.
The other thing is that I always worried about applicants who wanted to be approved to be permanent foster carers because it is a huge step to take, and the child is told this is her "forever family" and then it can go wrong. I always think it best to start of applying to short term a child and then it will give you an idea of how it feels to have a foster child in your home and get more information about how the system works. All Looked After Childrren have a Care Plan and that is either working to get the child back home, or have her adopted or permanently fostered. Very occasionally it can work out well as a child who is placed with carers on a short term basis willneed a permanent placement and none is available and the st carers are willing to consider permanency and these placements are very successful.
My advice is to contact the LA and get far more info. If you have birth children, most LAs like there to be a 2 year gap between a fostered child and a birth child, and a fostered child will do better as the youngest in the family. There are many other things to consider, so slow down, take it a step at a time. Taking a child into your home who has suffered trauma in the past i.e. sexual abuse, physical abuse and of course emotional abuse. These children can display challenging behaviour but many foster carers successfully care for these children and create a stable loving home for them.
Hope I haven't confused you too much. You can PM me if you like.