No problem at all that I can tell.. IF you have yet to spend money.. you might consider an "in-line" type inverter/charger.. These have both advantages and disadvantages.
For your setup this is how it would work, I will list pro/con later.
When you run the generator it will charge the batteries, Most of these units are very good 3-stage chargers and some can be programmed. The one I know best is no longer being made (Prosine 2.0 by Xantrex) but there are many fine ones out there by many companies, INCLUDING Xantrex. It will pass Generator power to the "house" (note. use a good inverter type generator like a Honda or Yahama please, not a contractor's earbusting model).
When the generator is shut down, it auto switches to inverter mode and the "house" never knew it lost power.. DO NOT RUN A/C during this period.
Basically, it is a a big honking UPS.
Pro's: Simpler hookup, you have only one set of connections to battery.
Con's. Though the combo-unit is likely lighter than an inverter PLUS a converter it is heavier than either of those by themselves.
Pro: less cable clutter
Con: if half of it dies, you go without till till it is fixed.
I rather like mine.. Tripped circuit breaker this evening (on 30 amp site) and did not even know it till I lost internet... The comptuer continued, the TV and DVR continued, cooking dinner continued (Steam cooker electrically heated) but I lost internet.. you see, all the stuff that worked was on the inverter, the MODEM, is not, it is on a mains only circuit (I did this for a reason) when I looked at the router (Which still had power) I noticed the status led flashing Orange
Flashing orange = No Modem
Steady Orange = Modem but no internet
BLUE = GOOD
On my router and thus I knew the modem lost power and "Vanished" Went out-reset and BLUE (ok) once again.
The "modem" is a Linksys WGA (Wireless game adapter) connects me to the park Wi-Fi.. looks to my system like a cable Modem but wi-fi instead of cable. kind of neat, Gives very good range with a minor modification.. i will finish the set up tomorrow when it is nicer out (Cold rain today, very hard on fingers).
Home was where I park it. but alas the.
2005 Damon Intruder 377 Alas declared a total loss
after a semi "nicked" it. Still have the radios
Kenwood TS-2000, ICOM ID-5100, ID-51A+2, ID-880 REF030C most times