Calling a temple

I would like to visit some temples that require a reservation in advance by telephone.

Could someone please help me correct the following phrases or add some new ones?
I don't want to say the wrong thing.

Thanks,
Kathryn

いつもお世話になっております。 (Can I say this to a temple???)

Kathrynと申します, アメリカからの観光客です。
お寺で仏像を見る許可を求めていただきたいんですが。。。

(if they agree)

よろしければ 今週のある日お寺に参拝させていただきたいんですが。。。

(if they agree)

何曜日が都合がいいでしょうか。

午前11時でよろしいですか。

本当に感謝しています。

(end the call)

失礼します。

Overall, what you have written is understandable. It may not be 100-percent native natural, but it gets the point across, and there's nothing impolite or offensive in it.

That said, I'd make a few changes.

1. 「いつもお世話になっております。 (Can I say this to a temple???)」

For your situation, いつもお世話になっております might be a little confusing, since it implies some sort of preexisting relationship between you and the other party, or at least the organizations you both belong to. So if you were to say いつもお世話になっております when speaking to someone at a temple, the impression they would get is that you, or someone from your company/school/family etc., has contacted the temple in the past, and so wondering about that might distract them from focusing on the current topic you want to discuss.

Personally, I'd probably open with 「すみません、うかがいたいことがあります」 ("Pardon me, but I'd like to ask a question").

2. 「よろしければ 今週のある日お寺に参拝させていただきたいんですが。。。」

I'm guessing you want to say you'd like to visit on a day this week, in the sense that any day this week is OK for you. In that case, just
「よろしければ 今週以内に参拝させていただきたいんですが大丈夫でしょうか？」
would be clearer.
「今週のある日」
sounds more like you already have a specific day in mind that you want to visit on.

Also, while it's not unusual in Japanese conversation to let the end of the sentence trail off and leave the question itself unspoken but implied, communicating like that can be difficult to do over the phone, where the other person can't read your facial expression and other non-verbal cues to see that you're done talking and waiting for a response. So to keep things clear, over the phone you're probably better off verbalizing the question, like with 「大丈夫でしょうか？」

3. 「何曜日が都合がいいでしょうか？」
This is understandable, but
「何曜日は都合がいいでしょうか？」
is more natural-sounding.

4. 「本当に感謝しています」
Again, this is understandable, but
「ありがとうございます」
is more natural. 「感謝する」 is more commonly used for large or long-term kindnesses that are already being/have been rendered. At this point the temple will have, at most, agreed to let you visit and set up a time, so 「本当に感謝しています。」 feels a little out of proportion, and might have the other party thinking that you're under the impression that they've agreed to some especially large accommodation, and perhaps something larger than they're actually offering.

Again, nothing in your original draft is impolite, and it's all understandable, but these changes will probably make the conversation a little smoother.

communication by phone is, in general, not easy.
simple is best.

こんにちは、ｘｘｘｘ寺ですか？
（はいそうです）
ｘｘ月ｘｘ日ｘｘ時ごろに参拝に行きたいのですが、ご都合はいかがですか？
（大丈夫です）
では、よろしくお願いします。
（はい、お待ちしています。）
*************************
ｘｘ月ｘｘ日ｘｘ時ごろに参拝に行きたいのですが、ご都合はいかがですか？
（この日は都合が悪いです）
では、いつがよいでしょうか？ or いつがよろしいでしょうか？
（ｘｘ月ｘｘ日なら大丈夫です）
では、ｘｘ月」ｘｘ日のｘｘ時ごろにお伺いします。
（おまちしています）
よろしくお願いします。

Kathryn,

Following is my suggestion - i am assuming you read Japanese well.

こんにちは。初めてお電話します。(Hello, first time calling.)
Kathrynと申します。アメリカからの観光客です。

お寺で仏像を拝観する予約をしたいのですが。
(I would like to make a booking to view the statue at the temple.) - since it is about viewing a specific statue, i would I keep this phrase 仏像 in it.

よろしければ、今週うかがいたいです。(If possible, i would like to visit this week.)
いつが都合がいいでしょうか。(When would be good for you?)
午前11時はいかがですか(What about 11?) - if you have a time you want to go.

ありがとうございます。XX日のYY時にうかがいます。- it would be proper to thank them, and confirm the date & time again just to be sure.

おうかがいするのを楽しみにしています。失礼いたします。(I am looking forward to visiting you. bye.)

Thanks for all the suggestions, much appreciated.
I will change my phrases accordingly to sound more natural.

Just hoping I will be able to understand their responses.
Sometimes it's a bit tricky over the phone.

Kathryn.