Can you give a link to the manufacturers page or a datasheet of the EEPROM.
The minimal 400 kHz is probably not in the datasheet. That would be very weird.
You may not power down a device and assume the I2C bus still works.
You found a trick with pullup resistors at the microcontroller, but that will not work in my opinion. Most chips have ESD protection diodes inside which make a current path from SDA and SCL to VCC.
You have to keep all the sensors powered, or you have to do a reset.
Can you measure the current that the sensors use ? Some sensors can be set in sleep mode and use only a few µA in sleep mode.
How much dynamic memory (SRAM) does the sketch have ? It is shown in the bottom part after compiling.
The maximum pulldown current for the I2C bus is specified at 3 mA.
When you have 5 pullup resistors of 5k // 10k // 10k // 4k7 // 4k7, that makes about 1k2 and 2.8 mA pulldown current.