Rebalancing for dollar's buying power at the time of the original deposit

I am not going to comment on how you will decide to rebalance. I am only going to talk about the frequency.

and then rebalanced once a week based on what that $10000 was able to buy on January 1st.

There is common advice to rebalance your portfolio. This is done to keep the balance between the sectors. If an investor wants to be 40% large company stocks, 30% small company stock, 20% bonds and 10% real estate as an example, then periodically they will sell some winners, and invest the proceeds in the sectors that didn't perform as well. This brings them back to their target percentages.

It is not unusual to see advice for an annual re-balancing, or a semi-annual, or even quarterly. But 52 times a year is a little much. I hope for your sake there are no transaction costs. Of course you could also set a percent range so that you weren't trying to sell investments just because the investment was just over the percent you set. Having a range means you might be able to skip some weeks.

It is also tough to do with a small investment. You may find that you need to be able to buy or sell fractions of a share, or you won't be able to get the percentages close. Some brokers will let you trade fractions of a share of stock, or fractions of an ETF.

If you want to do this then unless your inflation rate is extremely large, annual re-balancing should be enough.

Balancing is where you shift your investments around to maintain a particular percentage in each. It's not related to inflation or anything else - the value of the $ would be irrelevant.

Now, you could invest with the goal of hedging against inflation, and use inflation-linked securities as your "balance". Say, invest 70% Stocks / 20% Bonds / 10% TIPS (or more heavily in TIPS if you're really worried and don't mind the lack of growth). Then however often you decide to rebalance, and I echo mhoran's suggestion to do so not so frequently as weekly, you just keep those ratios the same.