Camille, there is absolutely no problem in your husband, a British Citizen, claiming the Public Funds to which he is entitled. So as regards Child Benefit, which absolutely must be claimed by your husband and certainly not you, he is entitled to claim that benefit and the fact that he is doing will have no impact upon your claim for ILR.
The Tax Credits, which need to be claimed jointly by a couple living together, given that your husband is not a person subject to immigration control, you are relying upon regulation 3(2) of The Tax Credits (Immigration) Regulations 2003, which reads (deleting the bits not relevant to you, to make it more readable) :-
Where one member of a married couple ....... is a person subject to immigration control, and the other member is not ...... -
(a) the calculation of the amount of tax credit under the Act, the Child Tax Credit Regulations and the Working Tax Credit Regulations (including any second adult element or other element in respect of, or determined by reference to, that person),
(b) the method of making (or proceeding with) a joint claim by the couple, and
(c) the method of payment of the tax credit,
shall ....... be determined in the same way as if that person were not subject to such control.
In other words, that joint claim shall be determined as if neither of you are subject to immigration control.
When those regulations were first made, back in 2003, I got my MP to ask a Parliamentary Written question to the Immigration Minister, namely :-
Mr. Simon: To ask the Secretary of State for the Home Department whether a claim under the terms of Regulation 3(2) of The Tax Credits (Immigration) Regulations 2003 (S.I. 2003, No. 653, by a couple including a person who is subject to immigration control will not be considered to be a breach of any restriction contained in a visa. [104441]
Beverley Hughes: The provisions of Regulation 3(2) of The Tax Credits (Immigration) Regulations 2003 will be fully taken into account when any changes are made to the list of public funds given at paragraph 6 of the Immigration Rules. In line with this regulation, receipt of tax credits by a couple, including one member who is a person subject to immigration control, will not be regarded as a breach of any condition of leave to enter or remain.
(Source : Hansard : 27 Mar 2003 : Column 345W)
So there you have it, a joint claim by a couple able to make a claim to Tax Credits because of regulation 3(2) "will not be regarded as a breach of any condition of leave to enter or remain".
So absolutely no need for you to worry about that Tax Credits claim affecting your ILR application.